2005 Acura RSX Not starting, then magically starting

The clue lies in that click you heard. If it was very faint, like a small relay clicking, that may not even be part of the starter circuit. You could have a neutral safety switch starting to fail or a bad connection in that part of the circuit. Burned contacts inside the ignition switch is another possibility. Those types of things generally start out as intermittent problems that gradually get worse over weeks or months.

While this isn't real common with your starter motor, what you described can also be caused by worn solenoid contacts inside it. That click will be real loud, ... Easy to hear from a long way off. That also starts out being intermittent. One day you'll have to cycle the ignition switch two or three times. By next month it might be a dozen times before the engine will crank.

The next time this happens, try shifting to neutral, then crank the engine. If that works, suspect the neutral safety switch.

This comment:

"If it was the starter then it wouldn't have started at all?"

is not correct. That is exactly what is meant by an intermittent problem. The potential worn contacts are inside the solenoid which is part of the starter motor. It can be replaced separately, but the time to do that, and hope it was done right, is usually not as good a value as simply replacing the entire starter with a warranty.

Before we even go that far to assume the problem is caused by the starter, lets wait for it to act up more often. Intermittent problems show up more and more often as they get worse. If it becomes a permanent problem, THEN we'll be able to do some tests to determine what is causing it.

Also, don't overlook the battery. If it is original, it has exceeded its expected life span of five years. Typically they fail suddenly and cause a no or slow-crank condition, but it is also possible for them to be just weak enough to not be able to engage the starter solenoid every time.