For a long time, from about book 2 to about book 8, we think Nynaeve is the strongest female channeler.  Supposedly, Rand chose her because (1)for her strength (2)because he trusts her as much as he does anyone. 

 

But by book nine, no less than three people have passed her up: Alivia, Sharina, and Talaan.  Rand may not know Sharina or Talaan, but Alivia seems to be the new strongest, and Rand seems to trust her by the time of the cleansing, so why not use her?

Because by the stage he knew of Alivia he'd already asked Nynaeve to do it.

 

Also... why not Nynaeve?

Mythologically, it ties in with Nynaeve's role as a healer.

Nynaeve ended up being strong enough and Alivia as a former damane would probably be better to have in the fight than sitting there anyway.

Rand says straight up she's the only woman he trusts to link with.  She was his village wisdom and he trusts her like a sister.  That trust is justified too, because unlike Egwene and Elayne she is most definitely not Aes Sedai first.  She's loyal to Rand.  She'd put him first.

 

When you're focusing everything you have on cleansing the source it helps to have absolute trust in the woman who's helping you.  Even if Alivia was a little stronger than Nynaeve, with the level of Sa'Angreal they were using he probably thought it didn't matter.

 

Well that's my theory anyway!

Yeah, everyone on this thread so far has brought up great points. All of it's true, and good reasons for why Nyn was the right choice.

 

Like Ark said, with all the hardware being used, and how powerful the channelers were, losing the actual power difference between whatever Alivia has that Nyn doesn't is probably a drop in the bucket. And Alivia is probably far more useful as a weapon then for what Rand was doing. And Nyn is one of the very few people Rand can really trust. But yeah....everyone already said that :P

I'm surprised no one else has mentioned it was the only surefire way for him to shut her up and quit pulling on the damn braid  ::)

Another reason for Nynaeve:

She knew how to link.  We do not know if Alivia knew how at that time.