Hmmm, so you have self resetting GFCI outlet? I didn't know they made those. On most of the ones I have seen you have to press the reset button.
GFCIs work by comparing the current between the hot and neutral legs, if the difference is greater than 0.005 (I think) amps, then the GFCI trips, because the missing current must be going through the ground circuit...which is bad.
Anyway, I think you are on the right track.
A lot of times one GFCI controls multiple outlets. Maybe test your outlets with the GFCI reset, then push the Test button and see which outlets turned off. If anything is plugged into these outlets, unplug it and see if the problem disappears.
If everything looks OK, then, like you said, try replacing it. I know from experience that these do fail from time to time. I just finished working on my Aunt's house because she had some outside outlets that didn't work. It turned out to be the GFCI.