You might be right it sounds like a solenoid. I would try to jumper the solenoid from the small wire from the button and the larger wire from the battery. If the solenoid actuates and the bike starts then it's most likely not the solenoid. If actuates and it does not start I would pull the solenoid apart and inspect the contacts and clean them with fine grade sandpaper or something like an ignition file that would work also. I think there is a relay in there somewhere between the button and the solenoid might look into that as well maybe that's the clicking you here.