What brand is your GFCI outlet? I installed a Leviton SmartLock when I redid a bathroom last year and the green LED stays lit on it. This particular outlet will not allow you to reset it if its miswired. It functions normally -- it passes power downstream and interrupts it when I hit the test button. I call the led a night-light now. I'm not worried about it any more.

I can explain it to you but I can't understand it for you.