For various reasons known best to him, I suspect each one appeals to RJ in someway as the love interest of his protagonist, but he couldn't settle on one.

Min functions well (for certain values of well) as a regular person who can relate to Rand and where he comes from and what he is going through. The problem with a love interest like that is you can't have her do much, otherwise she loses her normal qualities. If she grows in stature with the hero, she morphs into one of the other types.

Aviendha is kind of like a partner for Rand, in ways that Min is not. She is both a warrior and a channeler, with certain attributes that suggest her potential as a bad-ass and companion to Rand has been largely overlooked. Also, with her, RJ has given Rand an adversarial romantic relationship, which the other two could not credibly maintain in the same way. He can't turn to Min for comfort if she's as dangerous and ready to go toe to toe with him as Aviendha. Her coming from a society where woman hold the economic and a share of the political power enhances the equality point too. By any standard, Min is Rand's inferior, and by any official standard, Elayne is his superior. Their respective gender roles in their societies as well as the relative egalitarinism of the Two Rivers and the Aiel make them equals in many ways that the other two can't really match, however equal the personal footing on which they deal with him.

Elayne is the kind of love interest who is useful to the hero, with the political and social influence and power she brings with her. Obviously, a queen and Aes Sedai is a very socially fitting and practical partner for the Dragon Reborn. She also functions as another type of love interest - the reward. RJ would never be so sexist and chauvanist as to have a female character who exists solely to reward the hero for his accomplishments, but he's also too romantic to overlook that trope, either. So he subsumes it within characters with actual purpose, and makes the most reward-like love interest also the most significant and in-depth character of the three. While all three are described as attractive, and if pressed on the issue, Rand would no doubt dutifully profess an equal attraction to each, Elayne is the one most described in that way. Even Mat and Perrin note how good-looking she is in the explaining how she is not their type (noble and blonde, respectively - and fine arguments, too), as well as a number of apparently heterosexual female aquaintences. Both of her potential sister-wives seem to express an opinion in her superior looks compared to themselves - an insecurity neither shows any hint of about each other (Min wonders how she can possibly "compete with a beautiful golden-haired queen..." while Aviendha, in attempting to dissuade Rand from their future relationship, tries to divert him with a catalogue of Elayne's appearance). However evenly divided his affections, Elayne would seem to the girlfriend of choice for posting bragging pictures on Facebook, and her social circumstances are such that she would not only be okay with, but actually expect, to support her husband and their family. Given the lack of media and public relations issues in WoT, being the lacking-in-authority consort of a ruler has to be a pretty good deal if you can get it, and it ties in both the reward-relationship trope and the valuable-political-alliance aspect.

It is worth noting as well, that, in keeping with her status as the reward girlfriend, she falls for him with the least effort or even presence, on his part, and her criteria for gratification are almost appallingly low when seeing her as her own character and not an appendage to Rand (She reacts most positively to him asking her advice and reciprocating her feelings through the warder bond, as well as being willing to toss her a sloppy thirds pity-f*** ). Not only that, she appears to have superior, if redundant domestic skills to his fellow Two Rivers women, being cited as the best seamstress of the Wonder Girls, a good enough cook to draw company when it's her turn (as well as being the only one nice enough to stomach Nynaeve's cooking, so we know she's not high maintenance, either), impressively fertile and a former professional tightrope walker to boot. Personality aside, the woman has some impressive tangile attributes for a girlfriend (on the other hand, she's the only one of the three whose breasts have not had positive notation of their size, so no one's perfect).

In summation, by having three girlfriends RJ can create a character who can deal with Rand on a personal level and always be there for him without sacrificing her own development or growth (because she has none of significance) while still having love interests who can grow in power and ability, both personally and temporally. He can have a sympathetic & comfortable relationship, and challenging and adversarial relationship, and a more impersonal courtship narrative-reward. Given that they have had one personal conversation shown to the readers, it is hard to say much about the nature of his relationship with Elayne, so from an analytical perspective of her narrative role, it makes the most sense to look at her tangible attributes, rather than how she interacts as a character with him, hence the reward-issue. He gets the satisfaction of taming Aviendha's tiger and the ego-gratification of Elayne falling into his lap on her own volition and Min pursuing him sexually. RJ gets to write tragic, potentially unrequited longing from afar, misunderstanding-inspired squabbles, and cutesy, cuddly literary emetic flirting.

It would be just about impossible to write a single female character with whom Rand can have ALL those aspects of a relationship. Perrin, Mat and Egwene each get a relationship that has one or two of those (Perrin has the partner; Mat has the remote political & social prize; and Egwene the cute, slavishly devoted hanger-on), becaus you can't have just one love interest be all things to the character.