In my case the single click was the solenoid moving out as trigger current reached the starter, only to hit the dead spot on the ignition switch which cut the trigger signal, so the solenoid returned to rest. Never made it to the end where it engages the motor contacts.
It would start dozens of times in a row after an 'incident', only to fail the next time I tried it.
Ignition switch - $20 fix, 15 minutes of work.
