Here are a few tips when it comes to re-wiring an old lamp.
Use the existing wire to pull the new wire through. You need to join them in a way that does not create a lump in the wire.
Avoiding a lump
On the old pair of wires, cut one wire shorter than the other and strip off the insulation like you were going to make an electrical connection. Do the same on the new wires. Now put one long wire to one short wire and wrap them around each other making a continuous wire. Do the same with the other. Notice how the off-set keeps the wires away from each other and avoids a lump at the same time. This is actually a good trick for splicing any extension cord. One piece of tape between the two and a little tape around the outside and you have a smooth splice. If you were doing this to repair a wire, or lengthen one, solder the wires together to prevent pull apart.
Wire pull-out protection
Don't forget that you must tie a knot in the wire inside the base of the lamp to prevent the wire from being pulled by the cord. We call this a stress relief knot.
When you attach the plug, tie a knot inside the plug for the same stress relief which will protect the contacts from being broken. The best and smallest knot for inside a plug is called an Underwriter's Knot (for UL laboratory who approves electrical things). Follow the diagram, it's easy.
Polorized plugs and the "rib" on one of the two wires -- it's always there!
Although lamps don't often have what we call polarized plugs, plugs where one is wider than the other, if you are replacing a polarised plug in any appliance, the ribbed wire goes to the wide or neutral prong. On the other end of the wire, the non-ribbed wire goes to the switch. This assures that the circuit breaker is connected to the switch, not the rest of the appliance.