1999 Pontiac Grand Am Security Light/Won't Start

Computer problem
1999 Pontiac Grand Am Two Wheel Drive Automatic 115000 miles

We bought this car last year. The previous owner was a little old lady who took good care of the vehicle until she hit a deer with it, and rather than repairing it, she sold it "as is" to a dealer. We bought it "as is" and did the repairs to it. It did sit in a garage for about 8 months until we got the money together to do the minor repairs that had really nothing to do with the computer.

Every so often, I believe its happened three times, the car won't start, and the Security light comes on. The only way we have bypassed this problem is by disconnecting one of the battery cables and letting it sit for several hours unhooked. When we reconnect it after these several hours, it will start and we have no problems. I have been driving it for two weeks and it only just happened, but it happened right before I started driving it as well.

What could the problem be? I'd rather have an idea of what it could be, rather than pay someone to check everything.