F150 won't turn over

I have a 97 F150, starts when it feels like it, other times, won't turn over.
Had the starter replaced, solenoid, the battery is less than a year old, 12.75+/- volts.
Just had neutral safety switch replaced on Thursday, picked it up in the afternoon, drove it to work & home on Friday, everything seems fine. This morning, Sat., I went out to start it. Nothin'. Truck is won't turn over, just get the click of death when you turn the ignition.
Everything else works, lights, radio, etc.

This has been the pattern. Sometimes it will start okay for weeks, even months, then one day, click, click. when you turn the key. Take it someplace, they look at it, maybe replace something, sometimes it just works fine when a mechanic tries it.

The length of time that it starts okay is getting shorter and shorter, though.

This morning I had a slightly new twist.
When I opened the door to get in, there was a noise coming from the speakers, kind of like the static sound you get if you hold a cell phone too close to something with speakers. There was no power on in the truck, though. Key was still in my pocket, ignition off, radio off.

I suspect that whatever is causing this may have been from mice living in the dashboard. Had a problem with that last year, right before all this began. I just can't figure out just how it's related.

I'm starting to get a little desperate, if anyone has any ideas, I'm open to suggestions.