Team Spirit is contagious on your team, so what you have to do is make things fertile for infection.
Find someone who wants to be the spirit leader - someone who the students will follow because they're slightly (or more than slightly) kookie. It's especially effective if you can get one of the "cool" people to do this, so everyone sees it's cool to be spirit crazy.
Practice a few cheers at team meetings. Add motions, simple dance moves and not just verbal cheering. Add "marginally prohibited

" rhythm instruments (we beat on coolers, totes, etc.)
Get ALL the mentors to join in. Nothing is funnier than a bunch of 40 year old engineers and 60 year old grandmas doing hip-hop moves - the students will look forward to each cheer just to see us make fools of ourselves.
Don't just cheer when you do well - cheer just for the sake of cheering.
It took us a couple years to build up, but it's worth it - we definitely have more fun at competitions.