I'd go with most of the fellas above.
The solenoid does wear out.

Pull the starter motor out and pull the solenoid apart. Normally the terminals inside the solenoid are chunky copper. I've never asked if these copper bits are replacable, I normally just get a replacement reconditioned starter from the auto electrician.

These are what throws 12V to the motor part of the starter motor.

Also the old dirty terminals is a good one. Get a cup of CLR or good metal cleaner and soak the terminals in the mixture. That'll get all of the oxide off. Otherwise just give them a good scrub with a battery post cleaner, $5 at the local spares place.

The alternator is there just to charge, it won't affect starting, unless the terminals on the alternator are dirty and corroded.

Cheers,
Chris.