91 geo prizm won't start every couple of years but they cannot find anything wrong

I got stuck again for about the third or fourth time since I owned this car. I drove to an appointment and when I stopped at a store on the way home the car just would not start when I came out- I can barely hear a click- tried for about an hour off and on— then I called the towing service (again). The battery and alternator are fine and the mechanics are stumped- they let it run for about 45 minutes to get it hot but it still started right up for them over and over. Seems to happen every couple of years- each time no clue. I used to have a slightly bent key but had one made from code at the dealer once thinking that was the problem- but apparently not as it happened again- thanks to anyone for any clues or advice?

I love this car- it has sentimental value and even better gets 32 mpg still- not many other problems really.

It may be the electrical circuit which engages the starter solenoid, which allows battery power to go to the starter and run it and crank the engine. This circuit goes through, at least, three contact points. At a contact point, the electric power can: conduct from one contact to the other; or, it can spark arc from one contact to the other. The contact points where the electrical power arc across have a higher electrical resistance and a voltage drop. A capable mechanic, with an electrical multimeter, can take readings from the power source (a fuse) across the ignition switch, the park/neutral switch, the etc. switch, and the starter solenoid to see if there is a too-high resistance and voltage drop. It (almost) takes longer to describe it, than to do it.

Loose battery ground is what I will look for FIRST. Can give annoying and intermittent problems.

This vehicle is 16 years old.

Replace the positive and negative battery cables and clean all the connections before attaching.

The old Toyotas, before they were Prizms and before FWD used to do that because of a badly designed ground between the engine and the firewall. The firewall had a thick coat of paint on it and the ground wire was attached on top of the paint. The ground would work well enough until the stud got a little rusty after about 8 or 9 years. Yours probably goes to the front of a fender and may need the rust removed at the fender end. Probably not buy maybe.

BUT maybe