You can test it by plugging something else (like a clock) in there for a day instead of the FIOS box. GFCIs can go bad too, the outlet might need replacing. Is the FIOS box a recent addition or have you had it for a while?

Is it protecting something else downstream? Fairly common for a basement GFCI.

If you've got the battery in the FIOS box, you can try unplugging it for a bit. Let me explain what weedmeister is saying...the GFCI can also be wired such that other outlets (perhaps others in the basement, perhaps outside or in the garage) are also protected by it. When it's tripped, check other outlets and see if they are also out.

To answer:

The Fios box has been there for years.

My issue is that I won't have Internet, TV, phone if I unplug the box to plug in something else --- unless I can hook the box up to another outlet with an extension cord.

Yes, the battery is in the box, too; however, it can go hours before it trips, so not sure how long my battery lasts.

And, yes, there are other outlets (outside) that it protects and that are also out; but what does that tell me? That it could be those outlets causing the tripping?

From what you say, the first thing would be to figure out whether it's the box that trips the GFCI by unplugging it --- but it could go for hours/days before anything trips.

You could disconnect the load side wires from the GFCI for a while and see if it continues to trip. It usually takes a divide and eliminate possibilities approach to figure out what is tripping it. It could be something as simple as your sprinkler system in the yard has been knocked out of adjustment, and one sprinkler is soaking an outside receptacle when the wind blows just right. Or, by tripping the GFCI, and searching for everything that has lost power, you may find an unexpected piece of equipment on the circuit such as a sprinkler timer box with the cover left open, or a yard light that has a small cut in the UF cable feeding it.

Yes it could be water infiltrating into those outlets, for example. I had exactly that problem on my new house (I knew there was likely a problem due to the rather ugly slathering of caulk on the outside of the box). On easier way to diagnose it is to rewire things such that you have individual GFCI's at each outlet and move the feed to those other outlets to the LINE (unprotected) side of the existing GFCI.

I have one outside outlet (but on covered porch) and two inside hardwired lights that seem to be on the same wiring. But how do I know where the cause is?

OK, so I plugged in the Verizon box into a different outlet --- no tripping in over a day. Does that indicate that it's the Verizon box? The outlet it's currently plugged into doesn't have a GFCI.

As I said, there's at least one outside outlet and two inside hardwired lights on that tripping GFCI.

Anything is possible. In the FIOS installations I've seen the plug goes rather quickly into a "wall wart" style transformer (a box about the size of your TV remote control, white with a green light and then very small wires coming out the other side) and then to the Battery Backup (another white box with some lights/buttons and a door revealing the battery, usually mounted in the garage as well) and then to the ONT (big gray box on the outside of the house typically where the fiber comes in and connects to your phone lines and TV cables).

I would guess if the FIOS is tripping the GFCI, it would almost certainly be that "wall wart." Does it look like it has been damaged or wet or something. If it goes for a few more days with no tripping I'd call Verizon and see what they will do. Replacing the transformer might solve the problem.

If you wanted to test the box as the problem, you would need to plug it into a different GFCI to see if it trips that also.

Nothing looks like it got wet.

My reasoning was that if the outside outlet or the two inside lights were tripping the GFCI, then they should have done so within the last days. But that hasn't happened. Ever since I plugged the Fios boxes (both the wart and the main box) into a different outlet, my GFCI hasn't tripped.

I don't have another GFCI nearby to plug in the Fios to test it.

Temporary extension cord?

You get get a portable GFCI (looks like a short extension cord with a big wart on the end).

Hm, I could try that.

On the other hand, still nothing has tripped.

I'd gripe to Verizon.

Knowing them, they'll just come out and replace it all. Couldn't hurt. Might fix things.