Is it appropriate for a Christian boy to buy Hanukkah gifts for his Jewish girlfriend?

Yes, in fact, it is appropriate for him to show respect for all of her customs.
Hanukkah presents aren't a traditional part of Hanukkah anyway, so it really doesn't matter.

Answer:
According to Judaism, Jewish people should marry within their own faith. So the most appropriate thing for the boy to do would be to seek a girlfriend in his own religion. This can preclude a lot of later doubts and strife.