What you describe sounds more like a genetic algorithm than a neural netowork, but either way I think it's kind of silly. Both algorithmic paradigms work only if you very carefully design the starting conditions. And only then for certain types of problem. You'll need some sort of feedback for it to learn as well. I suppose you could do all of these things, but it would be much harder than implementing it with your own neural network (Bad dum tsssh) and almost certainly function an order of magnitude better.
That said, I want to see it done.
The perfect processor for this is a
Gumstix, which is a small linux computer with lots of convienent inputs and outputs. It's a peice of cake to interface to a robot controller, and perfectly FIRST legal to the best of my knowledge.
Go for it!