So it looks like a hard drive problem from the description in this thread. Fortunately gcortes has the solution.
I’m pretty sure I found the problem. I ran a disk verify from the Recovery Disk OS X Utilities. There were so many errors, it didn’t complete. I ran the disk repair and the verify again and it was happy. I didn’t reboot to test it as I was in the utility for reformat and reinstall the OS to give the Mac Mini to my nephew. With as many errors it had, I was surprised it ran at all.