Honeywell Digital "Round One" thermostat. I think it's a CT-2700 but I'll have to check. It's the one with simple two step programming.
The question is: Is there a battery in it? Something keeps the clock going when the furnace is off and there's no 24VAC, right? I don't think it could go too long on a capacitor alone. There's something on the PC board that could be a battery but I haven't probed it with a DMM yet.
It doesn't look replaceable--are you supposed to toss the stat when it goes bad? Ours is starting to have a flaky display where some segments of LCD numbers aren't working. Then again sometimes it's fine. I was thinking of taking a watch battery and (if I can solder to one without damage) stuffing that in there as a replacement. I can always hang a conventional cel outside temporarily as proof of concept. I mean the stat should normally be powered from the 24VAC so one would presume a bad battery would only cause loss of time when its off and not weird display artifacts at normal times but who knows.