Is the furnace blower operating when you hear this buzz? Can you hear the buzz immediately after adjusting the thermostat, before the blower noise is heard?
If the blower is operating, you may be hearing a resonant sound from either the furnace metal or a duct against that wall.
If you hear the buzzing immediately after thermostat adjustment, it may indicate a wiring issue.
I know it would drive me crazy in short order and I'd probably get a look into that space to see what the source is. It could be something as simple as a loose piece of wall insulation that vibrates when the blower engages. I'm guessing the tool for this would be an inspection camera for the least amount of intrusion. Some of these cameras have a small tube so a small hole can be drilled to access the wall cavity.
I'd also rule out the generator as a cause, using a cord to connect to shorepower or another generator.