I think that the Pattern tries to match up the Dragon with a suitable mate - someone who will satisfy him in every way. Considering he's the greatest man in the world, he needs a woman of equal prowess.

In the AOL, the Pattern might have tried Mierin initially. She seemed to have everything except that her personality was flawed, so that didn't work out. Ilyena seemed to have been the next 'ultimate woman'. We don't know much about her, but she was in no way some cheap barmaid that LTT was attracted to. As Ilyena Sunhair, she seemed to have considerable prowess as an Aes Sedai.

I don't think that any women of equal stature occur in the 3rd Age, which is why the Pattern clumped him with three women. Even if he'd been paired with the best of what the Age had to offer (probably someone like Elayne), he would have felt unfulfilled. Women like Ilyena don't exist anymore. We've seen Egwene and Min feeling unsubstantial when Lanfear appears - and that's just from her appearance.

The three women equalling Ilyena is an old theory, but I do think collectively they are Ilyena's essence - but definitely not her soul or ancestry etc.

Aviendha - passionate, aggressive, dashain Aiel, channeler, attractive
Elayne - educated, noble, channeler, tenacious, attractive
Min - empathy, goodwill, philosophy, uncomplicated, atractive

Ilyena was probably all of these in one.

I do think that AOL women (and men) are of much higher quality. We've only really got the Forsaken to go on, so that makes evidence tough, but ignoring their evil personalities they're remarkably accomplished people in general : almost all of them are attractive, physically strong and well educated. It seems that, like channeling, 3rd Age people are a level down in quality. They're a select group in many ways, so they're poor evidence - but Rand's visions of the AOL showed that the Aes Sedai in the Breaking were generally up there too. There really wasn't much for the Pattern to work with to suit up the DR with a mate.

I personally think that Nynaeve comes closest to AOL greatness (if you believe that Age to be superior : some readers don't). She seems to be an all-round accomplished woman, which is why Legendary Lan likes her so much.