RJ actually claimed, as partial inspiration, that he dated two women who knew about each other and were content with the situation. Also, the DR is a whole lot of man, and I guess one woman doesn't quite fit the bill for a time-riding spirit of anti-Destruction. But neither of these are intellectual answers...

Other posters, like DomA and Cannoli, have touched on the mythological allusions - and rightly so. But they didn't mention that the ladies also bear a resemblance (in that way) to the Norse 3 Sisters. They were a way of conceiving time (and fate?), where IIRC, one spun out the thread of lives, another wove them together, while the last one cut the thread. Those Wyrd Sisters represented Past, Present, and Future. You can see it in his women.

Aviendha is clearly supposed to represent the Past. She links him to his Old Blood, and he sees the AoL after meeting her. Elayne represents the Present, given her ties to the current political situation of the whole Randland map. Min represents the Future, because she (duh) sees the future.

It's also a writerly way of (a) creating tension, (b) giving characters more time, actions, breathing-room, (c) allowing our hero to live a varied life instead of just meeting one person (contrast SpiderMan, which started with Peter Parker's great love MJ), and (d) show us how different this world is compared to our own. IDk if it's more different because it happens or because they're all cool with it, or because it doesn't involve religious or social norms (exclude the Aiel, please). You can see a lot of reasons for it.

And, of course, many others have noted how they connect to Rand's various aspects, his development as a person, and the "convenience" of having these 3 gifted people around.

For myself, this love quadrangle only sort-of worked. But I've made several survey-style posts about Rand's 3 Women, so I'll leave off here...