Zombie Computer. Ok, I was going to give it one more shot before going down to JL and purchasing a replacement.
Took the cover off. Pulled everything out (one at a time) and then reseated it after blowing into any receptacle that might have dust in it. Unplugged and replugged all internal wiring. Disconnected all the external USB things - speaker, mouse, keyboard, hard drives. Switched from using HDMI to the monitor to VGA. Put in another new little battery. Put it back on 120V and connected it to a reliable transformer. Did everything short of dancing in front of it with a chicken over my head. (Think Voodoo.)
Powered it up. It runs again. Added one USB device at a time and rebooted after each one - still working. NO idea which thing was causing the problem, but it's running fine now. (Watched Guardians of the Galaxy on it last night, no probs!)
I feel so much better.... I really didn't realize how much losing that old computer was impacting my mood. Or, rather, not being competent enough to be able to fix it. (I used to, 20 years ago, put together my own machines from parts. But it was different them.)
SO, 450 pounds saved! Which is good, because the exchange rate has eaten that out of my budget anyway.