I write for fun. I write to be challenged. If I don’t think I can carry a story off, I’m much more likely to give it a try. Fan fiction is only a very, very small part of my total output, but it’s fun to do and gives me a chance to exercise my scribbling skills so they don’t get rusty, since fan fiction is easier to write than my own stuff. What do I like to write? Dramatic stories are everywhere, if you look for them and are willing to take the risk of diving in with your pen.
Also, there is a tradition in Judaism called Midrash, in which the Rabbis comment on the material in the Tanakh (Hebrew Bible). This material is interesting in that they expand on the Biblical stories, filling in gaps and expressing their own speculations and interpretations as to the motivations, etc. of the Biblical characters, intending to educate as they do. I feel that fan fiction can, should it choose to be so, be a sort of Midrash. Others created these characters, but each of us who reads about them or sees them will interpret what we read or see in our own way, and so we may each also comment on the original story in our own way through fan fiction, much as the Rabbis did with their Midrashim. Thus my fan fiction stories are often intended not only as entertainment but as commentaries on the original stories which inspired them.
Smart guys, those Rabbis ...