Great men like Rand shouldn't be paired up with great women. They'll have no time for one another, and the relationship will become sterile and will eventually fail.

Elayne's destiny is to rule Andor, and the queen doesn't have time for anything more than a plaything, nevermind a proper husband, nevermind the greatest man of all time. Aviendha is also distracted by Aiel politics and has obligations that greatly supercede her commitments to Rand. There's also an undeniable cultural difference, and while it may seem rather sweet that they're always stepping on one another's toes due to ignorance, in the end they'll end up irritating on another, and resent one another.

Min loves Rand a lot and she has no personal agenda to get in the way of their relationship. She's a lesser woman than his other two wives, but only politically, and her simple nature makes her his perfect mate. A man like Rand doesn't want to finish off a day fighting the Shadow and have to worry about the safety of the Andoran queen, or whether she can see him, or some wierd conclave of Wise Ones in the Waste that denies him from seeing Aviendha. He needs someone like Min who's often around, and is happy to read and just be with him.