You make a pick. Have a lot of fans answer. Then contact the staff of Fanpop. Of course, if the person who made the club still runs it (meaning they haven't left), then the process should be easier. If you're the owner, you go to the edit button on the top right of your spot.
1) Make sure people wants the icon and banner to be changed.
2) Have people suggests images for the banner and icon.
3) Make a pick using the images that were suggested (also make sure the question is along the lines of "Which image would you want the new icon to be?")
4) Set a due date to about a month or so, that way enough fans would answer it.
5) Go to the bottom of Fanpop. There will be blue words. "Contact Us" is on the right side. Send the link of the pick, the image, and a description.
..I'm going by what I've heard around the site. I'm not sure if any of the steps are wrong since I haven't contacted the staff yet.