Good questions, thanks.

I left out one important observation from the original message. All it takes to get the furnace to turn on in this situation is to manually lower the thermostat setting until it passes below the actual current temperature reading. At that point, there is an audible click, which I believe is the thermostat telling the furnace to turn off (which doesn't make sense, since it never turned on in the first like it should have). Then all I have to do is hit "Run Program" or raise the thermostat setting above the current room temp and the furnace will fire up like normal.

So, for some bizarre reason, when the temps inside the house drop quickly, the thermostat gets fooled. During milder weather, there's never a problem.

I assume that you have a Honeywell programmable tstat. Your problem is that it is an older model which does not have intelligent recovery. Meaning, that during the day you have it set at say 68, at night when you go to bed you have it set back to say 60 at 11 pm with it programmed to come on at 6 am for 68. What happens is that when you go to bed the stat shuts the furnace down until it reaches the setback temp [ +/- 2 degrees ] at 6 am the stat tells the furnace to come on and heat up to 68. Without intelligent recovery, the stat comes tells the furnace to come on at 6am, not to be at 68 at 6 am. The newer Honeywell models have the intelligent recovery feature which in your case would tell the make the house 68 at 6 am by cycling the furnace on/off throughout the niight to make it was 68 at 6am.