I was having a nice chat with Don Juan Demarco, the greatest lover that ever lived (or so he likes to be told, so I just humor him, even though I have no personal knowledge of that) and he brought up an interesting point in jest. He said he thinks Rand and Lan are brothers.
Though at first I scoffed a the very idea, I have come to the conclusion that it is quite possible. Before you stop reading this, proclaiming it rubbish, open your mind and come with me on a journey, a journey into the complex mind of Robert Jordan.
In RJ's books we have discovered that there are coincidences at every turn. The good guys don't die, everyone is related, everyone has had an affair with someone's mother, and so on (this has long been one of my complaints with the series, but I am told that it is all justified b/c the pattern is weaving itself so that Rand & Co. will have a chance of winning TG).
I believe it is quite possible that Rand and Lan are half-brothers, sharing a father. Malkier was up in the borderlands, about as far East as you can get, so it would not be totally out of the question for an Aeil to go up there. Why? Perhaps the Aiel and Malkier wanted to have some sort of peace treaty that no one knew about. Stranger things have certainly happened, and with Malikier being dead and the Aeil being tight lipped it is not outside the realm of possibility.
Ok, so Rand's father is in Malkier for some reason, and Lan's mother is a bored queen and needs a little excitement. Along comes this handsome, mysterious warrior man and she decides to have a little fling. Lan is the result, but of course, she can never tell anyone.
Years later this handsome man runs into Tigraine, and having a fetish for royalty, seduces her even though she is living life as a maiden of the spear. Rand is the result.
Why would anyone think this is true? Just look at the already odd relationships that Rand has with everyone. Galad is his half-brother; Lord Luc/Isam is his Uncle (and Lan's cousin--yet another oddity that makes this theory ring true); his adopted father could have possibly had some sort of relationship with his real mother's replacement, who just so happens to be the mother of one of the three girls he love; one of his girlfriends might be the daughter of Jain Farstrider; one of his best friends is married to the cousin of a queen. . .Rand's life is just bizarre, so why wouldn't RJ make Rand and Lan half-brothers? It would be a surprise to most people, yet it is completely possible since we know next to nothing of Rand's true lineage.
Also, have you ever noticed that Rand and Lan seem to have a kinship with one another? Lan has taken Rand under his wing and taught him all about sword forms. Rand is already a blademaster, though he is very very young. Coincidence? I think not. I think on a subconscious level Lan knows that Rand is his brother, and is helping him for that reason. Their father was a born swordsman who never even knew he had skill since he was an Aiel, but he passed the skill onto his two sons.