Once her father is dead she can be cured but you cannot convince her. Of the dialog options choosing "how are you feeling?" can lead to her choosing to accept a cure. If the player asks her directly if she wants a cure Serana will become defensive making it impossible to ever cure her. It is unknown as to when she will become unresponsive but it is confirmed that she will only consider a cure after her father is dead (Well dead for the last time)
It's like Kunigonde said, it's in the dialog options. You have to coax her into it, you can't just out and out tell her. Apparently she gets a little touchy when told what to do after you enter the Soul Cairn with her, help her stay alive, find her mother, convince the Dawnguard not to kill her (if you were on their side), and help her kill her long-term crazy father.
She can be convinced through dialog, though I dont remember exactly what I said, she ran off to Morthal and returned few days later, but nothing had really changed. She will still complain about sunlight but will be no longer able to make you a Vampire Lord.
Amulet of Articulation gained through Thieves Guild Quests make any speech check easier to win.
The dialogue options you're looking for go like this: Ask her how she's feeling after killing her father, then ask her if she's ever considered seeking out a cure, then when she asks you why, tell her that she could live her own life again. That's what I did, and it worked.
It's not just out and obvious, I think someone above mentioned the train of conversation you have to take to get there. You kind of have to make it seem like it was her idea, much like with any woman, for it to be a good idea, lol.
(Not trying to be sexist, so please don't take offense to my comment women gamers, I think y'all rock!)