Seemingly Random Intermittent Ignition Failure

I have a 1991 Honda Civic DX and it is doing something I can’t explain. Randomly the car will not even try to turn over. The ignition switch will click, the dash lights come on… but not even the slightest peep from the starter motor. If I put the vehicle in fifth gear and roll it, turning the engine over even half a rotation, the engine will fire right up. It started on a road trip and happened very sparsely, but has since become more common. What used to be 1 out of 20 is now maybe 1 out of 4 chance it doesn’t start. Every now and then it will go days without any problems, then seemingly randomly the ignition fails again for spurts.

I replaced the distributor cap, plugs, and rotor about 2 months ago. The ignition switch appears to be fine… any ideas?

It’s possible the click you hear is from a relay and not the starter solenoid.
If the assumption is made that the battery and battery cable ends are good the likely suspects would be the ignition switch, transmission safety switch, or the starter motor itself.

The small wire on the starter solenoid needs to be probed with a test light of VOM when the car is a no-start. If battery voltage is present then the starter motor is likely failing.
If voltage is not present then you have to go back up the string and consider a transmission safety switch or failing ignition switch. Those are easy to test and for what it’s worth, ignition switches have been a bit of a problem on Hondas for a long time.

They fail due to long term, high current flow through the switch contacts.