Aviendha is the most similar to Egwene, so perhaps that is why I think her aggressive personality is the perfect match for Rand's more passive one. Not only do they have a good balance, but she shows an inner dynamic fluidity - she is dangerous because she needs to be aware of the different levels of danger in the Aiel world and the Westlands, and because of it, she is the closest to being realistic in a way that women in Rand's land are not. Highborn women in the Westlands are used to the idea that they are valuable and protected by social mores - this doesn't apply to lower class women or those who seek other roles aside from committed motherhood and companionship.

Elayne is a very good counterpoint to Rand in some ways, but the fact that she couldn't arouse any romantic feelings in him until she declared it is a negative for me. Plus, they both are passive behind-the-scenes - she wants something from Rand that he is really not able to give. Whereas Aviendha inspired him to seek her out as a mate, Elayne had to take another strategy and be more coy about it, employing a number of strategies. This is neither good nor bad, since she is clearly more educated and intelligent than he is, but it does make them seem somewhat similar in overall mindset - despite the fact that Elayne was raised to lead, her strategy tends to favor using her perception to analyse those around her and then act, which means she often finds herself outmaneuvered because she can't strike the right balance of thinking and acting whereas Aviendha naturally had the latter and has the capacity to learn the former.

Min is the worst match for Rand IMHO because her personality was largely undermined by her position in the relationship she has with Rand. Furthermore, she has an inability to hold him to standards she has set and has become something of an enabler to the very mindset and actions that she finds abhorrent. I think the key thing about Min is that she has no courage or lacks the ability to convert her feelings into capability - the other women have been on a journey to learn skills, gain new strength and increase their capabilities throughout the series to become more powerful and live strongly. Min does not have that. While she is the most confident on the outside, she has no conviction and in the real world, her confidence is quickly brushed aside. As Aviendha comments (I'm paraphrasing from memory), "Doesn't (Min) realize that in the time it took for her to twirl those knives, (Aviendha) cut her cut her throat five times?" We see the same in both encounters with Semirhage - a legitimate threat appears, Rand is completely unable to draw upon his own abilities and both times, Min could do nothing more than superficial wounds. Min is unable to a be a warrior/killer, counselor/general or a resource besides constantly bending before Rand's whims. This is why she went from a character I liked to one I abhorred by the end of TGH, and I think there relationship sucks.