If the starter solenoid is operating properly and the car starts once in a while, it could be that the brushes in the starter motor have jammed and only contact intermittently. Those brushes spark furiously when power is applied to them and the brushes slowly erode away until they reach the end of their travel, this is usually when the wires attached to the brushes hit the brush holder.
Also the solenoid activates a switch that powers up the starter itself, maybe that is not working.
It all depends if the solenoid is working properly, if the click comes from somewhere else we can assume the starter is OK.
How many miles on this car?
I think this is the basic method of starters in operation, at least the ones I have monkeyed around with.