I'm sure my first reaction to Seaine was "Oh, not, not another random character that I don't care about!" but this quickly became a storyline that I like. The White Tower is fracturing even further and the events of Lord of Chaos have given a good insight into just how incompetent and petty Aes Sedai can sometimes be, so an example of cross-Ajah teamwork is even more welcome.
Seaine doesn't try to deny the existence of the Black Ajah, she tries to do something about it, even though she's terrified by the task. She realises she can't do this alone, even though she's promised to keep to keep this a secret. She approaches a friend from another Ajah, despite the current tensions between the Ajahs and that Pevara stopped talking her over a hundred years ago, and that she cannot be absolutely certain that Pevara will support her.
"You were very brave coming to me, Seaine. I've known Darkfriends to kill brothers, sisters, parents, to try hiding who they are and what they've done. I love you for it, but you were very brave indeed."
So Seaine is awesome. Pevara is, too. She says, "I cannot tell you how happy I am to see you [...] Can you forgive me?" and she doesn't try to stick her head in the sand about the existence of the Black Ajah, either.
And then they get down to the problem before them.
I also like Seaine and Pevara simply for their friendship. We see a lot of politics amongst the older Aes Sedai and not much else (and that makes sense, because we don't need to be privy to details of their personal lives. The books are long enough without that). But I'm glad we got this example of Aes Sedai being friends.