So my 67 New Yorker doesn't start sometimes. Most of the time it will. When it doesn't start I hear a clicking noise at the left side of the firewall.
Is typical for a starter relay to click when the battery is too weak to start the car, as the heavy current draw thru the solenoid to the starter causes the battery voltage to drop below the required voltage for the electromagnet in the relay. Relay snaps open, current to starter stops and battery voltage recovers quickly, causing the relay to connect again, repeating everything. Happens very quickly and causes rapid relay clicking.
Sometimes it will turn over for a second and then stop. If I leave it for a few hours it usually seems happy again.
This suggests environmental conditions might be at work? Cold engine, warm engine, ???
Battery is good, jump start doesn't make a difference, same in neutral and park. Could it be a bad starter relay? Or transmission lockout? Any other ideas?
If it was the neutral safety switch, you'd get only your dashboard idiot lights but zero response from the starter motor, as the NS switch breaks the circuit to the electromagnet in the starter relay. And you should be able to start it in neutral in that case anyway.
How have you determined the battery is good? I've seen weak batteries get passed by those modern computerized battery checkers at a parts store. (but yes, a jumpstart would seem to help)
Due to being intermittent, and without electrical diagnostic data of various things at the time the car won't start, here's what I would suspect:
excessive voltage drop thru ignition switch/circuit, causing the starter relay to flutter
starter motor that has a tight spot, that the battery can't overcome and drops down below the starter relay voltage dropout point
a battery that is borderline OK when the other problem doesn't arise, but can't overcome the other condition