when i installed an additional relay to activate my starter, and it has eliminated my starting problem for 18 months, so far, i concluded that my problem was not enough juice flowing through the ignition wiring / key switch to activate the solenoid. and i concluded this was the only problem. and since others have corrected what seems to be the same problem with new contacts in the starter i have modified my conclusions to this:
there is not enough juice flowing through the ignition wiring to activate the starter. the cause may by one or a combination of both of the following:
1. voltage drop in the ignition / switch wiring and
2. bad or worn contacts in the starter.
depending on how much you have of which one, may dertermine the best solution for your situation. i would assume on a rebuilt starter that the contacts would be better than average but who knows.
if you replace the contacts, they will work better and the vlotage drop, if any, may not be an issue any more. if you add a relay, you eliminate the voltage drop, if any, and with more juice the condition of the contacts is less critical, unless they are REALLY bad. i chose the relay because it was easier and cheaper and because i had already tried a good used starter. the relay works great.
measure the vlotage at the solenoid when you turn the key, this might tell you somthing. if the voltage is good, could it be the amps available that is the problem? i don't know enough about electricity to suggest anything else, but the relay works great. it eliminates any ignition wiring issues inside the car and steering column. the ignition wiring activates the new relay (it doesn't take much juice to do this) and the new relay activates the starter solenoid.
good luck.