We had the same issue last year. Our generator (Onan 5.5) cranked and ran perfectly every time prior to when it didn't. We were packing up to leave after a nice few days at the lake. Hit the starter and....CLICK! Hit it again, and....CLICK! So, I figured the house batteries were too far down to turn over the starter on the generator and proceeded to make the ride home with just the dash air. Got home, replaced the batteries, hit the start button.....CLICK!! CLICK!! Nothing... After diagnosing all the voltages at the new batteries and at the generator connection, and checking/cleaning/retightening all cable connections from batteries to generator, and manually jumping the terminals on the solenoid to see if THAT was the problem....I ultimately had a bad starter and had to replace it.
With all that said, check everything you can before you decide that it's the starter and then replace the starter motor. Not an easy task, but do-able in your driveway with an extra pair of hands and some patience. Actually removing the starter and replacing it is easy. Getting the generator out of the basement bin to get access to the starter ON THE BACK OF THE GENERATOR....well that's another job. If/when you decide to do the job, also know that the fuel line may be a tad too short to pull the generator all the way out to where you can get to the starter. I had to buy extra fuel line and add a coupler to lengthen it. You CAN just remove the fuel line from the fuel pump to give the extra room you need, but then you can't test the new starter until you reinstall the generator and THEN you'll have to pull the generator back out if something is wrong.
So, here's a great source to get parts for your Onan generator:
Onan Generator PartsAnd depending on which exact model of Onan generators you have, here's a parts manual:
Onan BGM Parts Manual