This theory is mostly a variation of a theory by Linda at the thirteenth depository but I disagree with her on one very important point. I would also like to  address a fair amount of additional evidence which she doesn't consider and so I want to record the whole thing here.

In short, I believe that the Aiel will have to return to the Way of the Leaf after the Last Battle.

Let's start from the beginning. At the end of tEoTW when Rand meets the Green Man,  the Green man says a number of interesting things to him.

Suddenly the Green Man spoke to Rand.
“Strange clothes you wear, Child of the Dragon. Has the Wheel turned so far? Do the People of the Dragon return to the First Covenant? But you wear a sword. That is neither now nor then.”
--tEoTW, ch 49

The first covenant is the term used in the AOL for the Way of the Leaf. BS mentioned on twitter that this scene is full of foreshadowings. I bet this is one of them.

                                BRANDON SANDERSON

Also, the foreshadowing in this sequence is brilliant. The Green Man speaks of things that become important thousands of pages away.

Next comes an Aiel prophecy mentioned by Urien in TGH

“Trollocs? Here?” Urien’s eyes brightened. “It is one of the signs the prophecies speak of. When the Trollocs come out of the Blight again, we will leave the Three-fold Land and take back our places of old.”
-TGH, ch 28

What were the Aiel places of old? In the AOL the Da’shain Aiel served the Aes Sedai and followed the Way of the Leaf. They also had an ability to amplify channeling using Singing along with the Ogier and the Nym. This seems to have been closely related to the Way of the Leaf.
Both the Ogier and the Nym were completely peaceful. The Ogier this side of the ocean largely remain so and they are the only ones who retain any Singing ability at the moment. It would be interesting to see if the Seanchan Ogier retained this ability as well. They appear to be a much more militant bunch.
But in any case, the core of what the Aiel were was the Way of the Leaf (just as it is ji'e'toh now). Rand sees this in his visions in Rhuidean

“Keep the Covenant, Jonai. If the Da’shain lose everything else, see they keep the Way of the Leaf. Promise me.”
“Of course, Aes Sedai,” he said, shocked. The Covenant was the Aiel, and the Aiel were the Covenant; to abandon the Way would be to abandon what they were.
--TSR, ch 26

One might reasonably suppose that becoming the servants of Aes Sedai might qualify as "taking back the Aiel places of old" but it's pretty clear that this is not the meaning of this prophecy. The Aiel attitudes toward Aes Sedai changed quite a bit and they will never go for it. Also, Egwene spent a good bit of time in ToM setting up a fairly elaborate structure of future relations between the White Tower and the Aiel. If they Aiel were going to become servants of the Aes Sedai none of this would be necessary and  would only waste highly valuable screen time.
Besides, there are several more clues pointing to the Aiel returning to the Way of the Leaf which I will address below.
In TDR, ch 34 Gaul tells Perrin of another Aiel prophecy

“Tear?” The Aiel sounded surprised. “Why . . . ? But it must be. Prophecy says when the Stone of Tear falls, we will leave the Three-fold Land at last.” That was the Aiel name for the Waste. “It says we will be changed, and find again what was ours, and was lost.”

One can argue that this has already happened but I don't think so. The Aiel haven't changed fundamentally yet. And neither have they found what they lost yet in any meaningful way.  Both things are yet to happen.

Another Aiel prophecy that Rand sees in his visions in Rhuidean say that

"he will come from Rhuidean at dawn, and tie you together with bonds you cannot break. He will take you back, and he will destroy you.”
--TSR, ch 25

so far Rand has not tied the Aiel with the bonds they can not break. Also, the part that he will destroy the Aiel can not be literal because another prophecy says that he will save a remnant of a remnant (more on that later). This means that the prophecy speaks of the destruction of what the Aiel are. This has not happened yet. I believe that the Aiel will be a part of the Peace of the Dragon where they will decide to accept the Way of the leaf after the LB. So while the decision to return to the Way will be made before the LB, the actual act of doing so will happen after the LB is over. This will be the bonds they can not break the prophecy speaks of.
Another prophecy from the The Karaethon Cycle that Moiraine mentions in TDR I think also points to this

He 'shall slay his people with the sword of peace, and destroy them with the leaf.'
--TDR, ch 6

Here  is where my interpretation differs substantially from Linda's. She thinks that the Aiel will have to convert to the Way before the LB and will not take part in the fighting. I think that's wrong for several reasons. First, there is a rather persistent theme in the series that every last man will be needed to fight the LB. Throwing away hundreds of thousands of excellent warriors before the LB is fought makes no sense. There is another prophecy indicating that this won't happen.

He shall spill out the blood of those who call themselves Aiel as water on sand, and he shall break them as dried twigs, yet the remnant of a remnant shall he save, and they shall live.
--TSR, ch 34

This means that Rand will use up most of the Aiel in the fighting. This hasn't happened yet.

Another possibly relevant prophecy

The unstained tower breaks and bends knee to the forgotten sign.
The seas rage, and stormclouds gather unseen.
Beyond the horizon, hidden fires swell, and serpents nestle in the bosom.
What was exalted is cast down; what was cast down is raised up.
Order burns to clear his path.

—The Prophecies of the Dragon
translation by Jeorad Manyard
Governor of the Province of Andor
for the High King, Artur Paendrag Tanreall

-LOC, Epilogue

This can refer to a number of things. The damane positions among the Seanchan  is one.  The adoption of the Way of the Leaf by the Aiel is another.  It would certainly turn things upside down in their society.

In TOM we get a couple more interesting clues. The first and the most important one is from Avi's visions in Rhuidean. They are often completely misunderstood and many people think that they show  Avi's  descendants being responsible for leading the Aiel on the road to oblivion. Nothing can be further from the truth.  Let's briefly look at the last two visions as they are clearly the most significant ones. The Oncala POV elicits a strong emotional reaction both from the readers and from Avi herself on account of Oncala so clearly being a horrible person.

"I don't want to go on," Aviendha said to the empty forest of glass.
The breeze had fallen still. Her comment was met with silence. Her tears had marked the dust by her feet, like fallen drops of rain.
"That . . . creature had no honor," she said. "She has ruined us."
--ToM, ch 49

But is this charge justified? Let's see what really happened in that scene.
Oncala doesn't say anything during the crucial negotiations and Hehyal does all the talking

"Ah," Queen Talana said. "The Taardad Aiel again. You still carry the spear, Oncala?"
Oncala folded her arms, but said nothing. She knew she was not good with people. When she spoke, insults were too common. Better to let the clan chief take the lead.

After the negotiations are over we learn that the idea to trick Talana was Hehyal's

"Is this right?" Hehyal asked as they walked, their spears surrounding them to keep away prying ears.
Oncala started. "It was your plan."

We also learn that Hehyal planned this for years.

How much could the Seanchan be trusted? Hehyal's agents had spent a great deal of time over the last decade seeding that very question among the great courts of the world. He was a wise man. Even before he had become chief, he had realized that this war could not be won by the Aiel alone. They needed these soft wetlanders.

So what does this all amount to?  While the reader hates Oncala for the despicable human being that she is, she is actually little more than a bystander in the whole scene.
She keeps her mouth very firmly shut throughout the entire negotiations with Talana, and we learn that Hehyal was the one who came up with the plan to trick Talana, stole the plans from the Seanchan, did all the talking with Talana and was actually trying to involve the Andorans in the war for years. Yet the reader assigns most if not all the blame for what happens to Oncala because it's her POV and she is so clearly wicked. This is a very clever misdirection of guilt by BS which is part of the reason I like this scene so much.

The last vision (Padra's POV) is even more important for understanding the true purpose of the visions. The readers tend to blame Rand and Avi's kids for the decision to go to war with the Seanchan. But again, is this justified?
Padra (and presumably the other kids) had nothing to do with the meeting. Padra had no clue what was going on there or what was on the agenda.  The agenda was clearly set the the clan chiefs. There are 11 of them and they all speak before any of Rand's kids say anything. The vote is 9 to 2 in favor of war before any of the kids say a word. They might share some blame but they are not the deciders. They are kids. Their opinions matter but the  real deciders are the clan chiefs. They are the ones driving the decision to attack the Seanchan. And  we see exactly what is driving them.

Alalved shook his head. "The spears grow restless. The Aiel are not meant to grow fat in lush lands, tending crops. We are warriors."
...
"Regardless," Ronam continued, "the spears clash; our people fight when they meet. It is our nature.
...
"The Aiel must have a purpose," Janduin said, nodding. "We are useless as we are, and we made no promise not to attack. It is a testament to our patience and respect for my father that we have waited this long."
...
"But war?" Padra said. "Is it right?" "I do not know," Ronam said softly. "We are Aiel. It is what we know
how to do."
Padra nodded, feeling more certain.
The Aiel would ride to war again. And there would be much honor in it.

The indicators of what went wrong are spread nice and  thick here including the very last line of the visions. The Aiel went to war because they needed a purpose as a people and their purpose has been to be warriors. This was going to happen eventually no matter what. That's the real lesson of the visions and not Avi's lack of parenting skills.
This is what has to change. The Aiel need a new purpose as a people.  They need something else other than war to give them honor. Returning to the Way of the Leaf and finding honor in peace as opposed to war fits right in here.
But when should this happen, before or after the LB? I already gave some arguments why I think this should happen after the LB.

The Nakomi encounter (TOM, ch 39) provides another. Whoever Nakomi is, she pushes Avi to start thinking of the need for the Aiel to change.

Interesting thoughts you raise, apprentice," Nakomi said, pouring the tea. "Our land here is called the Three-fold Land. Three-fold, for the three things it did to us. It punished us for sin. It tested our courage. It formed an anvil to shape us."
"The Three-fold Land makes us strong. So, by leaving it, we become weak."
"But if we had to come here to be forged into something of strength"
Nakomi said, "does that not suggest that the tests we were to face-in the
wetlands—were as dangerous as the Three-fold-land itself? So dangerous and difficult that we had to come here to prepare for them?"
...
"It seems to me," Nakomi said, "that by breaking our ancient oaths to do no violence, our people have gained great toh"
"Yes," Aviendha said. What did you do when your entire people had done something so awful? This realization was what had caused so many of the Aiel to be taken by the bleakness. They had thrown down their spears, or refused to remove the white of gai'shain, implying that their people had such great toh, it could never be met.
But they were wrong. The Aiel toh could be met—it had to be met. That was the purpose of serving the Car'a'carn, the representative of the ones to whom the Aiel had originally sworn their oaths.
"We will meet our toh" Aviendha said. "By fighting in the Last Battle."
The Aiel would therefore regain their honor. Once you paid toh, you forgot it. To remember a fault that had been repaid was arrogant. They would be finished. They could return and no longer feel shame for what had happened in the past. Aviendha nodded to herself.
"And so," Nakomi said, handing over a cup of tea, "the Three-fold Land was our punishment. We came here to grow so that we could meet our toh"
"Yes," Aviendha said. It felt clear to her.

There is no disagreement between Avi and Nakomi up to that point. They both think that by fighting in the LB the Aiel will meet their toh for abandoning the Way of the Leaf. Avi only goes astray when it comes to the question of  what the Aiel should do after the LB

"So, once we have fought for the Car'a'carn, we will have met that toh. And therefore will have no reason to be punished further. If that is the case, why would we return to this land? Would that not be like seeking more punishment, once toh is met?"
Aviendha froze. But no, that was silly. She did not want to argue with Nakomi on the point, but the Aiel belonged in the Three-fold Land.
"People of the Dragon," Nakomi said, sipping her tea. "That is what we
are, Serving the Dragon was the point behind everything we did. Our customs, our raids on each other, our harsh training . . . our very way of life"
"Yes," Aviendha said.
'And so," Nakomi said softly, "once Sightblinder is defeated, what is left for us? Perhaps this is why so many refused to follow the Car'a'carn. Because they worried at what it meant. Why continue the old ways? How do we find honor in raiding, in killing one another, if we are no longer pre-naring for such an important task? Why grow harder? For the sake of being hard itself?"

What Nakomi is saying that once the Aiel fight in the LB they will fulfill their purpose as warriors and will need something new. Moreover, by fighting in the LB the Aiel will finally meet their toh for abandoning the Way of the Leaf. But after toh is met one should stop doing what gained one toh in the first place. In this case it means that the Aiel have to go back to their state before they incurred toh, i.e. they need to change their warrior culture and go back to the Way of the Leaf.

However, the Way by itself is not a purpose. It's just a way of life. The Aiel need a more clearly defined purpose too. I'm fairly hazy on what exactly that will be. As I said before, I don't think the Aiel will become the servants of the Aes Sedai as they were in the AOL.
I do think that Rand will teach them the Song and they together with the Ogier will recover that part of their old functions. This is hinted at very early on

"And men cried out to the Creator, saying, O Light of the Heavens, Light of the World, let the Promised One be born of the mountain, according to the Prophecies, as he was in Ages past and will be in Ages to come. Let the Prince of the Morning sing to the land that green things will grow and the valleys give forth lambs. Let the arm of the Lord of the Dawn shelter us from the Dark, and the great sword of justice defend us. Let the Dragon ride again on the winds of time.

—from Charal Drianaan te Calamon,The Cycle of the Dragon, Author unknown, the Fourth Age"

--tEoTW, prologue

RJ formed the notion of Singing very early on. We have indications that LTT could do it. In tEoTW prologue he says to Ishy

"Have you the Voice, stranger? It will soon be time for the Singing, and here all are welcome to take part.”
--tEoTW, prologue

Rand now has full access to LTT's memories so he is primed for the task. It will likely be much needed too because even after the LB is over the world will be in a pretty ravaged state and Seed Singing will be very helpful in returning it to normalcy as the above passage indicates. It's from the fourth age, so from after the LB and presumably the author knew what he was writing about.

We learn from Avi's visions in Rhuidean that  part of the problem with the Peace of the Dragon in the visions is the absence of enforcement.

"This peace of the Dragon's will not last long, anyway," Alalved said. "Skirmishes between the nations are common, though none speak of them. The Car'a'carn required promises of the monarchs, but there is no enforcement. Many wetlanders cannot be held at their word, and I worry that the Seanchan will devour them while they squabble."
--ToM ch 49

It is possible that part of the new purpose of the Aiel will be enforcement of the Peace of the Dragon.
Given that the Aiel will return to the Way of the Leaf I'm not sure exactly how they will go about it. As some sort of unarmed UN peacekeepers perhaps?