I talked about the power of releasing different versions of your songs in my book, Six-Figure Musician. A "remix contest" is a great way to to this.
http://coleplante.com/tetrisremix/ has a great example of a contest Electronic Arts is doing for fans who want to remix the Tetris theme. Even if you don't remix music yourself, I suggest you sign up to see how they're running things.
Basically, it works like this...
1. You (and others) remix the song.
2. People vote for the remix they like best. (The "fan favorite" gets $1000.)
3. DJ Cole Plante picks seven finalists.
4. Cole picks a finalist who gets his version of the Tetris theme song used in EA's Tetris Blitz plus $3000.
Not exactly life-changing money, but a bragging rights for being in a legendary video game like Tetris is worth a lot, in my opinion. And even if you don't have a similar prize for remixing your music, don't underestimate how much being able to be part of your career will be worth to your fans.
An example of a remix that has been submitted...
Don't think your music would be great for a remix? You're wrong! But regardless, you can use the same concept to do a cover contest. Simply give people sheet music and/or charts instead of sound recordings to remix.
If you do something like this with your music, I'd love to hear from you! Reach out ot me via Twitter.