I had the same clicking sound. Battery good. Tried replacing starter. Then I found if you would move the wire and then have someone try to start it it would work. I think there was a short and it was grounding out the solenoid causing it kick out.

Crazy place for wire - underside of truck, right front - and Ford's lack of proper insulation/protection makes it a natural to rust. When you try to start the truck, you get a "click - click - click" sound like the solenoid is stuck. However, the hot wire connector to the starter was green with corrosion and wasn't making a good connection.