Theory: I propose that the dice in Mat's head are directly connected to the Eelfinn's memory-harvesting system.

In chapter 8 of Knife of Dreams, Mat remembers dying and comes to an unpleasant realization about his memories. He muses that his replacement memories had to have been hand-picked, so to speak, since there weren't any of anyone's childhood, like there might be if it was just a random assortment. Also, the memories tend to suit the characterization the *Finn gave him ("trickster, gambler, son of battles"). He wonders what the *Finn "do with memories, anyway".

We know from Moiraine that the Aelfinn "rummage around" in visitor's heads, though even she doesn't know why. The Eelfinn appear to do the same. Mat reaches the conclusion that

"Maybe they created some sort of link to any human who visted them, a link that allowed them to copy all of a man's memories after that right up to the moment he died." [Chapter 8, KoD]

Sounds pretty logical to me. But what could that link be? Ironically enough, it was Mat saying

"Having those foxes in his head was a thousand times worse than the dice however hard those dice rattled." [Chapter 8, KoD]

that made me think of the dice as that link.

After all, it's pretty obvious that they are in some way connected to his foretold future. They stop whenever something fateful happens- and they roll for battles (though never street brawls, interestingly enough) and for many things connected with Tuon. Very telling also is that we first hear of the dice rolling in The Fires of Heaven, chapter 42, after Mat has visited both the Aelfinn and the Eelfinn. Also, we've never heard of any other character having these fate-alarms in their head. Moiraine and Rand have both visited the Aelfinn doorway, but they only have other people in their heads, not nonliving objects. Partially because of that, I think that planting the dice was something that the Eelfinn did.

There are more reasons I think the Eelfinn did it rather than the Aelfinn. We know that there's some measure of collaboration between the Eelfinn and the Aelfinn, but they differ from each other in a number of ways. The Eelfinn seem more aggressive- they wear human-skin suspenders and took Mat's death as the price. Also, the very fact that they ask a price for their services makes them different from the Aelfinn- the customary "rummage through your memories at the door" isn't enough of a reward. They seem more likely to try to trick the customer (by planting a memory-collecting device in them) than the Aelfinn. Maybe as a result of those tricksy Eelfinn, foxes in WoT folklore are seen as much more sly than snakes.

In the same vein, they could actually have been fufilling their bargain, but in a sneaky way.

The actual text of Mat's first request is

'"I walk around with holes in my memory, holes in my life, and you stare at me like idiots. Well, if I had my way I would want those holes filled, but at least answers to my questions might fill some in my future. You have to answer--!"

"Done," one of the men growled, and Mat blinked.' [Chapter 24, The Shadow Rising]

He got the holes in his memory filled, sure enough, but he also asked for the holes in his life filled. What he technically asks for is the answer to his questions, but Eelfinn aren't the Aelfinn. They don't give answers, they grant wishes. So his wish for the holes in his life filled- that could be interpreted as a wish to understand his fate, as well as a restatement of the 'holes in my memory'. Indeed, he asks them to fill in the holes in his future as well as the holes in his past. Well, he already has a future, but he doesn't understand it, which is the problem. So they gave him a way to understand it- the dice. The dice tell him when something fateful occurs, keeping him informed of the future course of his life. But in order for them to know that something fateful has occured, they need to be able to scan his thread the way the *Finn can- thus, the dice are most likely directly linked to the Eelfinn.