It sounds like something is loose, making a connection sometime and sometime not. I think you are going to have to start off with a voltmeter and check voltage drops while you are only hearing that click and the starter not turning over. I will send you a wiring diagram, so place the voltmeter across the battery like you were checking battery voltage, it should read 12 volts or maybe a little more. Turn the key to start and when you only hear the click and see how much the voltmeter drops. That is how much voltage you are loosing somewhere, so that will be the amount we will be searching for. Let's say it drops from 12 volts to 8 volts. then you have a waisted 4 volt drop somewhere. Then move the positive test lead to the very next spot that isn't the same thing. That means, move the positive test lead to the copper cable exposed still right there at the cable end if it is one of those kind of battery terminal ends.
Then the next spot to move the positive lead to would be at the other end of the positive cable at the starter but on the copper cable unless the cable end is solid, then on the cable end connector. Now each time you move the test lead, you should still have 12 volts when you turn the key to start.
What we are looking for is that 4 volt drop. So, when you actually get to the starter with the positive lead on the big lug and get a reading, the next thing to do is start moving the ground test lead, toward ground while keeping the positive test lead right there on the lug at the starter. Once you get to the positive test lead on the big positive lead at the starter and the negative test lead actually on the metal where the ground cable is attached, move the ground lead right there to the starter casing, that will check the voltage lose from the starter casing to where it's bolted, incase there is dirt or paint on the mating surface or the bolts are loose.
One second on the wiring diagram.