Hmmm...minipcs for gaming have been so overdone it may be best to look elsewhere...
Heck, just look at 'Infinium Labs' and the Phantom (aptly named

).
However, you can do ALOT with computers and such, especially if you learn how to make use of the serial and parallel ports
I've seen all sorts of crazy ideas on the net you can get inspiration from (inspiration, not plagarism). You may be interested in doing something robotic, like a smart battle-bot or something (that would take a while to make though). I've seen a little robot controlled by a Ti-83 somewhere on tehinternet. Basically all you do is program the AI for it, however simple or complex. Maybe something to think about.
Main questions I need to ask are: a) When do you have to make this stuff by? b) How much are you willing to spend/How much junk do you have lying around? and c) What are rules and restrictions for these inventions (like, do they allow bits of gunpowder for a combustion chamber, etc?)