2002 Chevy Cavalier starting problem.

weak fuel pump relay, swap it with the A/C or defrost relay right next to it in the power dist.center under the hood

Unfortunately that wasn't the problem. That was the first thing that I thought also and it was a no go.

Check your ignition switch. When they get old they get real sensitive.

Check your ground wiring , ground to the battery and ground from battery to engine both are extremely important.