I'm starting to get the feeling that I should write one on here. I've written fanfics for years, there is a ton of things you can do with a setting or a cast of characters by throwing them into different time-lines, scenarios, introducing new plot-lines, exc. I had about 40 story ideas in the works at one point, and that is not an exaggeration. There are also cross-overs, re-imaginings, the ever-popular "let's ship these two" reasons for writing (which I am genuinely not ashamed to admit I've used) and so on.
Good fanfiction helps you to see old characters in a new light. It should still, ideally, feel like you're watching the same--or at least, a very similar person, but they're experiencing conflicts you wouldn't otherwise see... or you're seeing a new side of them that you wouldn't otherwise know about... so on. Exploration of an existing work can be a beautiful thing, really.