Does the car have VATS? The resistor in the key?
Next time it doesn't start, wait for a min 3-4 minutes, then try to start with the same key.
If it does start, and the same effect happens for later start-ups, I would probably change the ign lock cylinder due to the key socket being spread apart.
Inside the socket, there are two leaves rubbing on the key resistor. Every time the key is inserted, the leaves spread apart.
I had this happened to a friend's '91 Cadillac. Worked great after fixing.
If it still clicks, after hot starting or cold starting, I would clean the battery cable ends, both ends of the cable(s). Should have less than 0.3 volt drop from the battery post to the solenoid terminal when measured with a voltmeter.
Everett "OBJECTS IN THE MIRROR DISAPPEAR UPON RAPID ACCELERATION"