[NickR]

Sometimes the battery connections (and the ground connections) are good enough to run the minor electronics, but not good enough to start the car (the starter motor uses a lot more current).

Put a multimeter, set on volts, to measure the voltage across each connection while someone tries to start the car. If the connection is good, the volts should stay at zero. If the connection is bad, there will be a significant voltage drop (or rise, depending on which way round you have the leads).

Or, I wonder if it's the clutch position sensor? Try this: remove the thin "trigger wire" off the spade terminal on the starter motor. Make absolutely sure the car is in neutral. Then connect 12+ volts to that spade terminal (the easiest way to do this is to put a screwdriver across from the thich terminal for the wire from the positive battery terminal. If it cranks, the problem is the clutch position sensor, or ignition switch, or the wiring in between.