Well apparently cleaning the timer has fixed the problem. It's been about 2 weeks since I did that and since then the softener is back to a normal regen cycle.
I'm not exactly sure why cleaning the timer had that effect. I have some theories, but none of them are very good. Some of the...
I forgot to mention that I checked the toilets for leaks. This is a single-story house on a partially-finished basement, so any leaks large enough to account for these symptoms would have shown themselves long ago. There is only 1 outdoor spigot, and it's not leaking.
As for the electric...
It's about 15yrs old. I would say that it's been about 12-18 months since it started doing this. I took the timer apart and inspected the parts inside. The teeth on the discs look fine, but 1 of the pawls looks like it might be worn. There was a small amount of debris inside the control head...