If you were going to play dodge ball or any other game like that, you could consider mixing it up and saying that, 5 members (or insert reasonable number here) of each team (you could adjust this to size 5 of team XXX because it's a big team and 3 of YYYY because it's a small team) so that they play with people they don't know. You could also do games like pictionary, taboo, and charades, which are quieter smaller scale games (we had epic taboo at our hotel) and have mixed teams. I think you are best off to have any enforced activities (if you want to do any) earlier in the night (i.e. everyone plays dodge ball) and sort of open ended activities (play DDR, or taboo, or hand out and chat) later on, and then do a movie (if you want to) when it gets late and people are tired. Anything that takes organization will be harder to do the later it gets.
Glow sticks and anything that glow are a great idea (order a few weeks in advance and you can get those sorts of things from
Rhode Island Novelty or
Oriental Trading for quite cheap) and you know how much FIRST Teams like giveaways. You could also ask teams to bring a bunch of their buttons. If any team makes their own buttons, you could ask them to bring their button maker (people really do like this, even adults)
As far as food goes, I might try to avoid the pizza option just because I know that (and some of my team mates may actually kill me for this) people get tired of pizza by the time build and traveling are over. You might want to see if you can get some place to do a couple different kinds of pasta (baked ziti, pasta with meat sauce, pasta Alfredo) or a pasta bar. Whatever you do make sure you have a vegetarian option (meatless sauce, cheese pizza, veggie burgers/tofu dogs) and you might (if there are Jewish kids on any of the teams) want a Kosher option (something that doesn't mix meat and dairy, among other things). I might steer away from Chinese just because there are many people who don't like it, or make sure you have something else on hand. Have lots of chips/snacks and bowls of candy (cost co or BJs is helpful for this) and you may want to see if you can get some place to do a giant sheet cake (or a few) with the first logo or pictures of the robots on them.
As far as music goes you can do the DJ thing or just have people bring Cd's but someone should be making sure the music is appropriate. Try for not over done FIRST music, but don't play to much of one genre at a time. (And yeah, you're gunna want the Cotton Eyed Joe anyways)
We also used to do a raffle, although I don't know if that would work for you. Many places are willing to donate prizes and you could either charge ($1 for 4 tickets) and give the money to a local FLL team, an FRC team that has financial trouble, use it to pay for the party somewhat, or donate it to the local food shelf. Or you could give everyone ten tickets or give tickets for winning things or whatever.
Be sure to ask teams if any kids parents can help you. Someone may have an uncle whose a DJ or a parent who caters or something.
I hope some of this has helped and that it has not been in excess. I used to plan parties for the nonprofit theatre I worked for. (Yeah, I'm 17, but I've had experience) so if you have any questions you can PM, as I always did pretty well.
