If you are going to replace the gfci , read the labels carefully. There has been a manufacturing change:
The "line" terminals are now at the top and the "load" terminals at the bottom near the ground (green) . As pointed out by an above respondent the polarity on these is critical for proper operation.
If it does turn out the fault is on the "load" side of the receptacle; at least you have a good idea where to look as they are the devices that are off when the gfci trips.
When using a core with a "polarized" 2 prong end ( like a hair dryer) the gfci my not trip if no path to ground exists. For example if the dryer falls in a plastic tub of water with plastic pipes.