After my last lamp project, my new rule is to ALWAYS rewire old lamps. A replacement socket and a extension chord to use for the chord are cheap and easy to install that it's good peace-of-mind to replace them. I will keep the existing wiring with newer lamps, where there a reasonable confidence in the quality level.
this is my last lamp project
DOCUMENT, DOCUMENT, DOCUMENT
Before doing any dissasembly make sure to take a bunch of pictures so that you can tell how things go back together. Even a fairly simple project like this can have a bunch of washers and screws that you wont remember where they go when you finally get around to putting it all back together.
This should be fairly obvious but make sure that the lamp is unplugged before doing ouching any of the wiring. To disasemble a lamp start by gently preying the socket loose and then unscewing the existing wiring. pull the existing wire out. With the wire pulled out, unscrew the bottom nut and take the lamp apart.
SAVE ALL YOUR PIECES.
Save all your pieces, even the things that you don't think that you'll need again. Keep all your pieces in a covered container like a tupperware bowel so that you don't look all those little screws.
With the lamp taken apart, take the time to clean all the non-electrical pieces. I start with simple soap and water. this will get rid of the dust and crud that has built up over the years. let everything dry completely before moving on. For the glass use rubbing alcohol to clean it. Over the years, I've rescued many lamps from the garbage that all they needed was a good cleaning.