I have a GFCI outlet that will run fine for days than trip. Attempts
to reset fail, but the next day it will reset OK then stay on for about
a week. I replaced the first GFCI outlet with a new one with the same
results.

The only loads on the circuit when it fails are two Limelight
nightlights (not a lot of draw).

Any ideas what might be causing this to happen and/or how to correct?