Dodge neon speedometer just stopped working. What would cause it to do that? And if the cord connecting it came undone or something then how could I fix it myself?

On most model year Neon's, the Vehicle Speed Sensor (VSS) receives power (8 volts on the Orange wire) and ground (on the light Blue wire) from the Power Control Module (PCM) and provides vehicle speed data to both the PCM and the instrument cluster (on the White/Orange wire). Power is only applied with the key in the "ON" position. If you have power and ground, you likely have a failed VSS.
How does the car shift? Like normal?

. Check for codes.

. Check for poor connections, particularly on the VSS and to the PCM.

. Replace the VSS, if the problem persists.

. Replace the instrument panel, if the speedometer still doesn't work.

You did not state what year your Neon is, but it is a typical problem of the first generation Neons. Check the neon websites. Instrument cluster is not making good contact with the wiring harness plug in behind. Remove top dash plastic panel, no screws, just pops off, remove instument cluster, held in place with 4 screws and it pulls out. You see what resembles 2 plugs at the back, 8 or 10 prongs each. Twist the prongs slightly to give them a better connection and re-install. Sometimes just hitting the top of the dash with your fist will work temporarily.