2000 won't start

My sister in law has a 2000 5.4L 2wd Excursion that suddenly will not start. When it happened to her the other day she opened the hood and checked the battery terminals for "tightness" and wiggled the wires around the battery and the thing started up. Then last night we were going out with her and the thing would not start in her driveway. I then checked the wires and we could not get it to start. I loosened the negative wires from the battery terminal (there's a wing nut holding them to the clamp) and retightened them and it started. So we're all thinking it was a bad connection. Drove to a restaurant, ate, then came back out and it would not start. No matter what we tried it would not crank. We ended up having to call a tow truck. The battery is newer and fully charged. The terminal clamps are all tight and clean. The starter relay is clicking. The starter solenoid switch on the inner fender is clicking. The headlights remain bright and do not dim at all when trying to start it. I tried jumping the main high voltage terminals on the solenoid and nothing happened. She had the starter replaced several years ago when it was doing the ripping sound thing, when the motor starts spinning before it engages the flywheel. She thinks they installed a remanufactured starter. Would it likely be the starter? With snow and slush on the ground at the restaurant I could not crawl under it to take a look. This was a Florida truck and is now getting used to Wisconsin winter for the first time. Any help is appreciated.

If the starter solenoid is clicking, it is either a weak battery, bad connection or a bad solenoid.

The battery is fine. We even added a jump start battery to it and nothing happened. I did jumper the solenoid and nothing happened as well. It's been a long time since I did that, you'd usually see some sparks when you connected the poles and then the starter would engage. And the thing did start for us once and the starter spun as fast as normal with no signs of a weak battery. I'm wondering if there could be a dead spot in the starter motor.

It was the starter and yes the monster is up and running fine. Thanks.