Update: It turns out I did have one alternate component option - the motherboard had an onboard graphics chipset, with a monitor connector that had been taped over by the manufacturer. After removing the graphics card and connecting the monitor to the onboard graphics, the PC started successfully.
So, it looks like the graphics card was the culprit. It's a GeForce 8600GT. If I replace it with a GeForce 210, which seems to be the current entry model, should I get comparable performance from it?
Thanks for the memory suggestion - it was a possibility I hadn't thought of.