When I do fan art it is usually a copy of an image so that I can pick up ideas or techniques from the artist, or it is a redesign of someone's clothes or armor. That's one of the ways I got into comics in the first place -- as a child, I used to trace Betty and Veronica from Archie Comics and then redesign their outfits. Comics and fashion go way back for me!
I have only done illustrations of my own composition for other webcomic artists, though this hasn't happened very often. I want to try this again to challenge myself. I am going to work up some ideas for concept. Because my concept isn't that clear yet, I won't discuss it. I will say that it will be for Dragon Age.
Last, I will share some bizarre fan art I made for a friend. We make absurd fan art for each other, and this masterpiece came to mind after reading her community challenge blog with the theme of "heroic." Things quickly got silly. I secretly blame Lord of the Rings for the strange monster.
(Read her comic here, a sci-fi romance called Bio-Revelation.)