Man I fought that on a 3016 a few weeks back. It would click, and click, and sometimes not. If everything worked right and it slammed in just right it really turned the starter over well. By the way John Deere had a lot of issues with this series of Onans in their 316,318, and 420 Tractors. They even have a "Starter Improvement Kit" to help that situation.
Their Kit basically takes juice from the Positive wire and thru a rectifier, it also is grounded well on tractor somewhere back to that rectifier. Then the Blue wire that excites the solenoid is attached to that boosted rectifier, and a wire then leads out of the rectifier to the solenoid with the boosted juice. The issues is that after the key switch sends the juice thru all of the (corroded -probably) safety switches there is a real voltage drop, so it doesn't want to start.
Note: if the voltage is high enough and that solenoid slams well enough, the magnetism on that end of the solenoid pulls the other end toward where the wires go in. That movement then pulls a lever back toward the wires and if it is a strong enough pull it engages with the starter. I had one checked a starter/generator rebuilder, and he tested it several different ways and said "that is a real good starter" On the tractor it wasn't working, so pulled it back out and was working well. Switched solenoids between starters and it worked. The original solenoid would not move that lever far enough to engage. 3 bolts to change solenoid.
Deere also has had issues of the same kind on their Kawasaki Engines, and the new series Briggs engines. Their everyday price is right at $40.
Bill Moyer