It turns out that I was too hasty. The usual grey apple screen with the spinner came up first, then it eventually went away to the bluish screen. But by that point the hard drive kept making retry noises and it never got to the login window. So I disassembled the case again, and removed the drive assembly according to this disassembly guide. I cleaned and reseated the drive connector, but this didn't improve the situation. I still don't get to a login window.
I think the next step is to pick up a replacement 2.5" IDE drive and swap the drive out. I'm not 100% certain that the problem is actually the drive, but I think it's worth a try. Unfortunately, the most recent backup I made of the Mini is perhaps a couple of months old.