We found it! It does have to do with the heating system. We haven't heard it before, because we haven't needed the heater on at night, or there was more background noise. When the furnace needs to ignite, there is a noticeable "click" from the thermostat in the hallway, a heavier "click" in response from the cold air return on the other side of the bedroom, followed by another "click" of recognition from the thermostat. Every time. The sound of the heater fan blowing isn't all that noticeable, so I didn't connect the clicking noise directly to the furnace cycles.
I think there are other clicks involved as well as all this digital electronic stuff communicates with each other, but I haven't sorted them all out yet.
But we have identified the source.
The solution is simple: turn on a fan for background noise; or suck-it-up, get over it, and get used to it. I think it was the mystery keeping me awake more than the actual clicks.
