Repairing the wiring of a three-socket lamp involves removing the lamp shade, light bulb and harp section to get to the socket shell. Remove the socket, loosen screws holding wires in place and remove the wires. Connect the wires to a new three-way socket and put lamp parts back together. Make sure the lamp is turned off and unplugged from its power source before taking it apart.
Wiring a three-socket lamp is a simple process, but it does require a socket that is new or in good working condition. If uncertain what type of three-way socket to use, bring the old socket to a hardware or home improvement store and ask for help with finding a replacement.
Unscrew the light bulb and set it aside. The metal bracket that holds the light bulb in place is called a harp. Remove it to get to the socket. The socket displays instructions to press down at a specific area, and pressing with one's thumb releases the socket from its base.
Lifting the socket exposes the wires. Use a screwdriver to loosen the terminal screws. Take note of the location of the wires in relation to the terminal screws. A colored or black wire is usually the "hot" wire, with a white wire neutral. Remove the wires and set the old socket aside. Attach the wire leads to the new three-way socket terminals in the exact configuration as the old socket.