Two things I especially like about Anya as a character: 

1. First, her love of money. Of course, there’s a lot of comic potential (and comic actuality) in Anya’s avarice, but there is also a deeper and more moving aspect to it (which I think the show touches on at one point, though I can’t cite an episode). Anya has spent a thousand years as a vengeance demon, exacting terrible revenges upon faithless men. The straightforwardness and impersonality of monetary exchange must seem like a breath of fresh air in comparison. “But I have their money. Who cares what kind of day they have?”
    
2. Second, her “strangely literal” tendencies. Again, we’re initially led to believe that this is a consequence of her being “newly human” and hence somewhat unsure of how to interact with other people. However, in the episode where we are shown her pre-demon life with (also pre-demon) Olaf, Anya-the-human is just as literalistic. That’s just how she is, whether human, demon or human again.