If the connectors are tight to the battery and/or starter lugs.. There is a loose connection between where the wire is crimped to the connector either on the battery wires or a wire on the starter.. Corrosion and a loose connection would cause those type of problems...

It's possible that the ignition (key) switch is at fault. You don't mention just what wires you are jiggling. No matter what wires you touch, it could be a coincidence that it starts on the next try. Next time, go through the motions of jiggling wires without touching any of them (otherwise saying wait a few minutes and/it pop the hood) and see if it will start. If it does, then it's the switch.

Replace the battery cables. Jiggling the cables tells you what?

"I move the cables and then it starts."

Well, there ya go. Either the cable ends are loose or a cable or cables need replaced. The next time it happens, move one cable and see if that works. If it doesn't, it's the other cable.

Why does it always happen first thing in the morning though? It never happens during the day or at night after the car has been running all day.

Battery, alternator has been tested and is fine. I have a brand new starter. The cables ands connections were just checked and cleaned by a mechanic. This always happens in the morning and it will start after jiggling the cables a bit. I have never needed a jump to start the car. I have a 2006 Honda Civic 2006 Sedan with 214,000 miles on it.