Definitely a one-owner name. Pity, because it sounds good. And then there's the connotation of Insanely Optimistic, which isn't actually that fair to the actual character, but public opinion isn't going to change.
I like
Polly very much as a stand-alone name, and
Anna not at all. Pollyanne might work, like Sallyanne or
Maryanne, but I'd rather have just
Polly on its own.