When my '94 FLSTN sits for a while, a week or two, attempting to start it results in the solenoid just clicking ~ after pushing the start button several times, it will connect and start. After the first start, no problem for a day or two.
I have replaced the starter & solenoid, new battery, on a Battery Tender, cleaned battery connections. Thought I had found the problem in the rollers in the ignition switch ~ replacing them helped, but not consistently...
Don't know where else to go.....
???

 

This really sounds like a ground problem, Check ALL the grounds very closely and the integrity of the ground leads.