+1 on the battery as the fault instead of the starter.

The starter is, by design, a heavy load on the system. If the battery is weak, the starter will be the first to show it. Slow cranking, that sort of thing. Eventually, the starter can give a click and then nothing, or several clicks but no start. When the battery weakens, it's ability to deliver current is diminished. If the starter demands 50 amps, whatever, but only 40 can be delivered, the starter will click (starter relay, really) and the demand will cause the voltage to fall so the relay turns off and the voltage rises again so the click happens again. This repeats until you let go of the key.