It could be a bad receptacle or it could have a ground fault on the load side. Turn off the breaker and disconnect any wires connected to the load side and restore power. Then see if it will reset. If not bad receptacle, if it does you have a ground fault somewhere downstream.

GFCIs go bad and that's what you could be experiencing.

To expand on what Groundrod said, GFCIs can also be used to protect other devices - what's called "downstream." If any of those devices have a fault, the GFCI will not reset. Turn off the breaker and pull the GFCI out of the wall. There will be two wires (hot and neutral, often black or red and white although not always) going into the top of the GFCI, providing power. If there are two more wires (hot and neutral, not a bare or green ground wire) connected at the bottom, disconnect those wires. Cap those off with a couple of wire nuts. Push everything back into the box and screw the GFCI it in place. Then turn the power back on and see if the GFCI resets. If it does, then it one of the devices downstream causing the problem. If not, the GFCI is bad.

If there is only one pair of wires feeding the GFCI, there is (obviously) nothing downstream and the GFCI needs to be replaced.

"neutral connection/impedance upstream, maybe even at the main panel?"

Downstream maybe, but not upstream so much (though newer GFCIs prevent incorrect installation).

A GFCI device measures the hot and neutral current downstream (LOAD terminals of the receptacle) and from the receptacle in the device (if it has one) for balance. Any difference greater than ~0.005 amps trips the GFCI.

Newer models of GFCI receptacles will not reset if the incoming power is incorrectly hooked to the LOAD terminals instead of the correct LINE terminals.

If the device has been replaced in the last few years it could be incorrectly wired.
If it WAS operating and now is not this is very unlikely.

Moisture (bathrooms and exterior) can cause GFCIs to fail.
They are not very ruggedly built (no coating on the electronics inside) so moisture leads to corrosion and failure.

In most cases simply replacing them is the easiest step to fixing the problem.

If the new one works the old one was bad.
If the new one does not work you need to look further.

Thanks for the added input, folks. The switch is fine now...apparently when it rains and moisture gets into one of the outside, covered switches, it affects the inside GFCIs too. When it dried out outside, the inside one started working again. Weird.

The water is providing a path for current to leak and the GFCI is sensing the leakage.

The leakage is only ~0.005 amps (5 milliamps) to trip a GFCI).

In addition, the GFCI induces a small signal on the hot/neutral so that a ground-to-neutral fault is also detected.