The off-season, or as we like to call it, the "pre-season," can be just/almost as important as the build/competition season. The most important thing that your team can do during the pre-season is get active in your community. Go out and fund-raise, mentor some FLL teams, or march in a few parades. The more your community knows about you, the better off you'll be later on. You could also construct a cart or a pit area.
But to get back to your request for a technical project...
This past year MARS (2614) held its first-ever internal robot competition. Basically the team split in half and worked on a challenge similar (in spirit at least
) to a FIRST game. We called it Cube Capture and it came complete with a rules manual and a game animation
http://marsfirst.org/sites/default/f..._Animation.wmv 
.
The goal was not only to develop new technologies (the judging rubric/point values were weighted in such a way as to encourage team members to get better at what we perceived as our weaknesses) but also to train new and returning students so that when the seniors left for college it wouldn't be as hard on the team. The robots that were built for the competition weren't all that pretty (since they were built on very tight budgets) but they didn't have to be. Everyone had fun and learned a lot and it prepared the team for the 2011 season.
It doesn't really matter
what you do over the summer, just so long as you do SOMETHING to keep the team active and learning.
Good Luck!
-Luke