The bet originates in my 'Gawyn will kill Rand' theory, which evolved quite a bit before TOM was published. (I came up with it in April 2010, and by November I had written in Moiraine.)
The official wording of the bet, if I recall, was that Gawyn would mortally wound Rand, in the sense that he will give Rand a wound that should kill Rand, and would kill Rand given the passage of, say, more than a few minutes. This wound may or may not actually kill Rand. Either way, Rand will die shortly thereafter.
Here's the deal: I believe that Gawyn will play a major role in Rand's death for a few reasons:
1. Foreshadowing language. There is a bit between Gawyn and Egwene in LOC where their language gets quite flowery for RJ, and in retrospect there are some very blatant connections to Rand's foreshadowed death in that passage. They talk of betrayal, and the key bit is when Egwene says "Betray? Gawyn Trakand, that word fits you as darkness fits the sun." We have a twice-dawning day, foreshadowing of 'the shadow at noon', Jesus parallels, and all sorts of parallels to suggest that Rand's death will coincide with some sort of abnormal darkness, maybe even the earth's rotation being reversed. (Some think that unlikely, and it might be, but there is plenty of foreshadowing that actually could make it work.)
2. Arthurian parallels and Jesus parallels (among others). Gawyn is most blatantly Arthurian, a parallel to Gawain, whose power (in some legends) waxed and waned with the rising and setting of the sun. He was, theoretically, strongest at noon. (That is also when the Dark One's powers are weakest, and the safest time for Rand to die, not to mention the fact that either the summer solstice passed on the day Gawyn fought the Bloodknives, or we're close to it. The chronology is not nailed down in Brandon's books.) Gawyn's parentage also makes for a Mordred parallel (Morgase+Damodred) and the odd relationships of the Arthurian characters (mainly the Andoran royal family which Elaida Foretold will be the key to the Last Battle) are a point in that direction as well. Mordred is a bit of a mystery parallel. He has three clear ones on the dark side—Moridin, Mordeth, and Demandred—and there are some of the light side as well. Gawyn is one of them, but Moiraine is too. That's why I don't think Gawyn is likely to kill Rand outright. But he is foreshadowed for a betrayal, and if it's not for the same reason as we thought before, then perhaps for some other reason. Egwene is a Guinevere parallel, so there is a touch of betrayal there too, and Egwene's Accepted test, where Rand begs her to kill him to save him from being turned to the Shadow, seems to buttress her place in the saga of Rand's death—how it comes about, specifically. She has a major part to play there.
3. The importance of swordfighting to Rand's character. Now, this is where some would say that Galad is a Mordred parallel, being a Damodred, but Morgase is not his mother, and RJ said in an interview that Galad and Rand are not going to have a swordfight. Gawyn is also a blademaster now, and he's looking for a sword, since he didn't take Hammar's sword when he defeated him in TSR. It just so happens that, since Rand has acquired Justice, he has an extra sword—the sword of Laman Damodred, the Oathbreaker himself. That sword has Gawyn's name all over it. And Gawyn has three Bloodrings. What will he do with them?
As for the sword being important to Rand's character, there are a few points to make on that:
1. Twice and twice he was marked, herons and dragons. This doesn't necessarily mean anything, of course, but the symbology is strong. Why was it important for Rand to become a blademaster at all? Just because RJ thought swordfighting was cool? I don't think so, because he contrasted the subject quite often. In fact, going back to the Arthurian parallels dominating the Andoran royals (Rand included), Henri Haslin was Master of the Sword for the Queen's Guards until Rahvin came along and got rid of him, at which point he became an alcoholic. Rand hired him and brought him to the Black Tower to teach the sword to Asha'man. Rand and Taim had a fight over that, since Taim sent him away. Rand told him to bring him back, because the Asha'man needed to learn the sword in case they couldn't channel. As for that fight...
2. In the first chapter of LOC (same book where we meet Haslin, get the Gawyn betrayal foreshadowing, first clear prophecies on Rand's death, etc.), Rand is practicing the sword against five men at once—the five best he could find. He gives them loads of gold to hit him. After Sulin pays up, Bashere asks him "Why?" And then to demonstrate the point, he throws a dagger at Rand's head, which Rand stops with a weave of Air. Naturally all of Rahvin's toadies (Arymilla, Elena, etc.) were quite offended and made as if to apprehend Lord Bashere, but Rand stopped it, and returned Bashere's question. With all that out of the way, Rand said the same thing he said to Taim—that you couldn't always count on being able to channel. All of that appears to be foreshadowing for Rand's capture at the end of the book (he killed a few Warders with his bare hands), but alongside the other foreshadowing—including Min's viewing (in LOC) that Perrin has to be there for Rand not once, but twice, or it will be 'very bad' for Rand—it seems more like endgame stuff to me. Brandon confirmed that Perrin's second time is yet to come. (Some thought it was Perrin's Dragonmount scene in TOM). I think it will be in Tel'aran'rhiod after Rand's death, at which point Rand's soul will be a target for Slayer (who also ties into the Andoran royal blood, and Lan's blood as well).
3. The blademaster thing is important to Rand quite literally from day one, when Tam pulls the old trunk out from under the bed, and when he gets wounded, Rand ends up having to use it. Tam gives it to him the next day, and on the way to Baerlon, Lan (the best swordfighter in the world, according to RJ) starts teaching him how to use it. Just before Tam gives him the sword, Rand has a dream where we see his first Lews Therin memory, and there's a great deal of debate as to whether Rand's quickness in picking up the sword didn't have anything to do with subconscious memories. Throughout the series, Rand has had two heron-mark swords, and Justice. When he started channeling, he made a heron-mark blade from the Power. He's dueled three blademasters so far, and led another to his death (Sammael). Also interesting is that there is a pattern connecting Rand's battles with blademasters to his encounters with Mordred parallels. He hasn't encountered Demandred at all, but this is still interesting:
—Rand battles Turak at Falme, and then encounters Ishamael/Moridin/Moerad in the sky, earning himself his first unhealable wound. (And giving Ishamael a pretty bad one too.)
—Rand battles Be'lal, a blademaster, and they both fight with swords made from saidin. Ishamael shows up, and Rand kills him.
—Rand spars with Toram Riatin, a blademaster, and then a bubble of evil strikes the camp, distracting Rand so that Toram can strike the 'winning' blow. In the chaos, Fain/Mordeth nicks Rand's already unhealable wound, making it doubly unhealable.
As for how Rand is supposed to pull off a decent swordfight with one hand...well, he is ta'veren. And also, more foreshadowing:
New Spring 26 - When to Surrender said:
Lan shook his head slightly. "He was better. But he thought I was finished, with only one arm. He never understood. You surrender after you're dead."
Moiraine nodded. Surrender after you are dead. Yes.
As I said before, I think that Moiraine is also a strong Mordred parallel, and so I think it quite likely she will play a major role in his death. There is plenty of evidence to suggest that Rand's death is crucial to his success. The Egwene Accepted test is one of those moments, certainly. Rand wanted her to kill him to save him from being turned to the Shadow. Gawyn might be able to pull that off, but there's a big possibility that balefire is required, in which case I expect Moiraine to be the one to deliver the final blow. It's slightly problematic, but only because Rand's body is supposed to be burned. But Egwene's dream of Logain stepping over Rand's body suggests that it might be an Illusion weave—it fell apart when she touched it, which is precisely the weakness of an Illusion weave. If Rand is balefired, and they still think it's important for the world to know he is dead, then they might put an Illusion weave on Gawyn's body and burn it. Egwene had a dream where Gawyn came to a fork in the road. Down one path, they married; down the other path, not. Down one path, he died old and in bed; down the other, he died young in violence. She doesn't know which choice leads to which path, but Gawyn makes a choice. (In case you're wondering, I don't think the Logain dream means Rand will fake his death for several reasons, including four prophecies stating that he will, in fact, die.)
So, Gawyn and Egwene will probably be heavily involved, in essentially predictable ways. It may be that Gawyn betrays Egwene; it may be that Egwene gives the orders. It may be that they set out to kill Rand for something like the prevention of him breaking the seals; it may be because Moridin's influence somehow takes hold and Rand actually needs to be taken out for everyone's safety. It may be that Egwene actually becomes aware that her Accepted test scenario is coming to pass, through a dream or some other means. It may be that she realizes he has to die to sever a link between himself and the Shadow. But again, it may be that Gawyn acts on his own, perhaps reading more into her idle conversation (i.e. 'Will no one rid me of this tiresome priest?') than she intends. He has promised to obey her, with one exception: he has to be able to protect her. Fans are really divided on whether Gawyn being involved in Rand's death is disappointing (to them) because he is a schmuck, or because they don't think he will do anything dishonorable, but I really think that whatever RJ has planned will make sense.
As for Moiraine's role, and balefire, there's foreshadowing for that too, way back in TEOTW:
The Eye of the World 13 - Choices said:
"We haven't forgotten," Perrin said. "It's just – " Still not raising her voice, the Aes Sedai treated him as she had the gleeman.
"Is that the way you all feel? You are all eager to run off to Illian and forget about Trollocs, and Halfmen, and Draghkar?" She ran her eyes over them – that stony glint playing against the everyday tone of voice made Rand uneasy – but she gave no one a chance to speak. "The Dark One is after you three, one or all, and if I let you go running off wherever you want to so, he will take you. Whatever the Dark One wants, I oppose, so hear this and know it true. Before I let the Dark One have you, I will destroy you myself."
It was her voice, so matter-of-fact, that convinced Rand. The Aes Sedai would do exactly what she said, if she thought it was necessary. He had a hard time sleeping that night, and he was not the only one. Even the gleeman did not begin snoring till long after the last coals died. For once, Moiraine offered no help.
I've written several scenarios as to how this might turn out, working in Alivia's role in various ways, but I won't pretend to know every detail of how it will go down. I just think that Gawyn will get a killing blow in, and that Moiraine will show up in the nick of time to save Rand from mundane death and make sure the Dark One can't get his soul. As she said later in TEOTW:
The Eye of the World 41 - Old Friends, and New Threats said:
Perrin raised his head to look at her, and Rand gave a start. The irises of his friend's eyes were more yellow than brown. With his shaggy hair and the intensity of his gaze, there was something about him... Rand could not grasp it enough to make it out.
Perrin spoke with a soft flatness that gave his words more weight than if he had shouted. "We can't stop it alive, either, now can we?"
And as Moridin himself said in TGS:
The Gathering Storm 15 - A Place to Begin said:
So some of the others really were dead. Balefire was the key. But how had Moridin gotten into Rand's dreams? Rand set wards each night.