Hmmm like 60hz sound like a transformer? If you take the panel off you'll see a black motor with a small flywheel on it, does that spin up at all? Try to spin it manually see if the 60hz hum changes, I wonder if the motor is fubar, or the capacitor for it.
Though what you could also do is when it's doing it, look for a flashing LED on the circuit board, you may have to open up the blower compartment and hold down the safety switch, though on my furnace there's a small window where I can see the LED so don't have to take the cover off. Basically it will flash in a certain pattern like long long short short short, record that and somewhere on the panel cover or other area should be an error code table. That's pretty much how I figured out my issue.
There is also a 24v transformer in the furnace for the thermostat, maybe it's shorted out somewhere so when the thermostat relay clicks it's just shorting out the transformer and it's humming. Though that would require quite a wire fault...
I guess try to figure out exactly where the hum comes from and go from there.
I'm not an expert mind you, I just figure these things out as I go, but this is my educated guess.