1998 Dodge Intrepid Starting Problem

1998 Dodge Intrepid Starting Problem:
When I try to start my Intrepid, I hear a single click and no start. Sometimes, it finally starts; sometimes it doesn't.

I have taken the car to my Dodge dealer (twice) and it always starts for them. They have checked the system without finding any problems. The dealer mechanic says that he has checked all elements of the system, including connections. He says the problem is either with the starter or the ignition, but none of us wants to replace parts without determining the problem. They say they have tested both the starter and the ignition, and both test OK. They have no other ideas without seeing the problem occur with them.

Note: I replaced the battery, cable and plugs during normal maintenance in June of this year (prior to having this problem).

Any suggestions on how I might proceed?

starter solenoid is getting weak. Your best bet, if the battery cables are good, clean and tight and your grounds are good, clean & tight as well, is to replace your starter.