A: What can't vinyl tape fix?
The button board had been acting flaky for awhile, and often to set the clock I'd just open it up and short the accumulator pin to adjust the time. It turned out that the material between the buttons and the board itself had dried out over the years, thus squishing instead of depressing as expected. I tore it out and used some tape in its place.
(to Antony: this is the same one I modified earlier. Why buy a new clock if I can just fix up what I have?)