Do you have a multimeter? What are the engine #'s?
Generally a click at the solenoid is from low voltage, alot of solenoids are replaced because people think they are bad.
Okay, test starter stud coming from battery should be 12+ volts. Disconnect the blue wire and place positive probe(Volts dc on multimeter) to it and negative probe to good ground while trying to normally start mower. What voltages do you get? I would test this a few times.
I mistyped that, should have been from solenoid stud(battery side) 12+V. Sorry for the confusion.
Does the starter appear to be engaging and disengaging properly, not hanging up(bendix) when switched off?
I get about 6.6 volts on the blue wire when it just clicks. What do you mean by battery side or starter side? This was measured on the terminal where the blue wire connects to on the starter/solenoid.