Let’s approach the problem systematically. Sorry for the long post.
We’ll discuss the time-related aspect of this in a moment.
We know* the ignition switch, when turned, sends a 12 volt signal that eventually ends at the starter solenoid, powering it and engaging the starter. In a '66 mustang this is a bigger wire that leads through the neutral safety switch (NSS) and then to the starter. In a modern car with an ‘immobilizer’ system, it is more likely like this: (from Autozone.com)
The starting, or cranking system consists of the battery, starter motor, ignition switch and related wiring. These components are connected electrically. When the ignition switch is turned to the START position (and the theft protection module recognizes the key code, as equipped) battery voltage is applied to the starter solenoid (through the theft deterrent relay, as equipped) S terminal and the solenoid windings are energized. This causes the plunger to move the shift lever, which engages the pinion with the engine flywheel ring gear. The plunger also closes the solenoid contacts, applying battery voltage to the starter motor, which cranks the engine.
We then see this info for testing the starter:
Disable the ignition system by unplugging the coil pack. Verify that the vehicle will not start.
Connect a voltmeter between the positive terminal of the battery and the starter B+ circuit.
Turn the ignition key to the START position and note the voltage on the meter.
If voltage reads 0.5 volts or more, there is high resistance in the starter cables or the cable ground, repair as necessary. If the voltage reading is ok proceed to the next step.
Connect a voltmeter between the positive terminal of the battery and the starter M circuit.
Turn the ignition key to the START position and note the voltage on the meter.
If voltage reads 0.5 volts or more, there is high resistance in the starter. Repair or replace the starter as necessary.
OK, if the starter and solenoid are OK, you should then trace the starter solenoid wire from the ignition switch, through the NSS, to the theft-deterrent relay. Then from there to the starter, after checking the relay itself (just the electromechanical relay, not the electronics).
OK, what’s one minute have to do with it? Heat. Something is warming up to a critical point and failing. A can of freeze spray might be valuable here. My personal bet is either the NSS, or maybe the relay itself. Jumper wires can help isolate it…
In one car I owned, the ring gear sensor (a magnetic sensor that detects the starter teeth on the ring gear at the back of the engine, to allow the motor controller know the engine’s RPMs) was flaky. At a certain temperature, it would fail - shutting the engine off as if the key was switched off. Waiting about 1 hour allowed it to cool down, and the engine ran again. For a while. Replacing the sensor solved the problem.
Really, you need a wiring diagram of the starter circuit to properly trace the start signal from the switch to the starter. Do that and you’ll find the problem…when it is happening only, of course.
I found a diagram on the AutoZone.com web site, it is attached below.
As you can see, the theft-deterrent module supplies a ground to the relay.
OK, get out the multimeter and have fun!
*maybe