Check all your connections, like you are going over someone else's work, for tightness and clean connections. Verify that your battery is at full charge. Get a load test done on it or borrow a friend's load tester. Verify the amount of voltage at the battery and the amount at the battery with the engine running, as already recommended. With the car off it should be at least 12.5 volts. Running should be 13.5 to 14.5 volts.
Another test is to put an inductive Ammeter on the cable going to the starter and reading how many amps the starter is drawing when activated. You would have to look up the spec for your car but I would guess about 180 amps should be normal. If it is drawing too many amps that indicates a problem with the starter, either improperly meshing with the flywheel or just the starter itself having a flaw. If amperage is low then it may have a problem getting all it needs.
Use a volt/ohm meter to verify the solenoid is not internally shorted.
Steve
Check the starter for proper installation depth into the bellhousing, depending on how it mounts. Sometimes there are shims that are needed to assure proper mesh with the teeth on the flywheel. Improper shimming can cause binding or clashing between the two.
After all that if you still have not found the problem get the wiring diagram for this circuit of your car and start checking at the ignition lock cylinder for proper voltage and then check the system following the course of flow of voltage from the ignition lock cylinder to the starter for proper voltage.
