(Disclaimer: I'm not an electrician, but I have rewired a couple fans and some lamps and haven't burnt the house down yet. Always consult a professional before trying something like this. Any rewiring you do at your own risk, OK? I'm just showing how I did things. Your mileage may vary, yada yada yada.)
Tools needed - camera, flat head screwdriver, adjustable wrench, pliers or wirecutters, razor blade, resealable plastic bag, replacement electrical cord or an extra extension cord you're not using, and possibly some brass cleaner or very fine steel wool.
First, unplug the lamp from the wall! Remove the shade, harp, and lightbulb. Next, look at the base of the lamp. A lot of these old lamps have either heavy felt or cork or cardboard glued to the edges of the base.
(I used adhesive-backed cork shelf liner for the base of this lamp.)
Take note of what is used on your lamp. Carefully get a razor blade or something sharp under the edge and peel back the base material, exposing the bottom of the lamp base. Start shooting photos as you go to document how things go back together.
There should be a brass nut holding the base and the threaded brass rod together.
(If the lamp wobbles sometimes it's because this nut isn't tight as it should be.) The cord typically runs up through the base and through a threaded brass rod up to the socket the bulb screws into. Use your adjustable wrench or pliers to loosen the nut on the base and unscrew it.
There may be a washer with the nut, and there's usually a heavy weight there to keep the lamp from being top heavy. Take photos as you disassemble so you can figure out in what order these pieces go back together. Put these small parts that come off the lamp into a ziplock bag so you don't lose these pieces.
Now try to pull up on the brass tube and the cord. If the lamp body is put together from several pieces be sure you may have to unscrew things at the top end of the brass tube too.
This lamp had a threaded piece at the base of the socket which screwed on to the top of the brass (black painted) tube.
Be sure to take photos of what order the pieces are stacked. When I take a lamp apart, I keep the parts in order they were removed and take a photo of that.
(By the way, if some of the pieces are tarnished or rusted, you might want to plan to shine them up or even paint them to give the lamp a fresh look when you put it all back together.)
You should be able to pull up on the lightbulb socket now. Look under the socket and see if it is attached to the brass rod with another threaded nut. If there's a nut there somewhere, try to unscrew it by hand or use the needlenose pliers or adjustable wrench to loosen the nut. Sometimes the bulb socket is just held in place without a nut.
The wiring feeds up through the base of the socket. Look closely at the socket and you'll probably see that it's really two or three pieces. This socket has an outer brass sleeve, an inner cardboard sleeve, and then the piece the wires attach to.
You should be able to grip the ends of the socket and turn your wrists counter clockwise about a quarter turn to separate the pieces of the socket. Now you should be able to see that the wiring is curled around a screw, which holds the wire in place. Usually these screws are flat head screws. You don't have to remove the screw completely, just unscrew both screws enough to allow the wires to come loose.
Now you should be able to slide the wiring down through the brass rod and remove it. Again, take photos of every step you make, and bag up the parts. ( If you are worried that you won't be able to tell which part goes where, use a separate ziplock bag for each part and number the bags in the order you have removed the part. Taking photos, making notes, and bagging parts is especially important if you will be shining up parts or painting parts before reassembly, because the longer you wait before putting things back together, the likelier you are to forget how you're supposed to put it back together.)
If you do lose a nut, or if the bulb socket is messed up, you can buy replacement parts like this at a hardware store. Take the old one with you and match it up with a new one. Take the old wire with you too and try to match it up with a replacement cord of the same length and color. (I have used an extension cord as a replacement, just cut off the end you don't need and use your wire cutters or a razor blade to split the wire into two parts about an inch or an inch and a half long. Carefully expose about a half inch of wire by peeling back the outer coating. Now the ends of the wire should be ready to screw back onto the socket.)
At this point, if you need to do some cleaning, polishing or painting, take care of that. (Use fine steel wool and some brass cleaner to shine up tarnished or rusted metal pieces.) I hit the brass pieces with some brass cleaner and they shined up reasonably well.
The most tarnished pieces were the hoop and its base, and they'll be covered up by a lampshade anyway, so I focus my polishing efforts the most on parts that will be exposed. Sometimes you'll find that brass or other metal is just too tarnished to shine. If that's the problem, try scuffing the parts with fine steel wool then prime them with spray primer, and after that's dried, pick a paint color or metallic color that you want to use to cover up the tarnished pieces.
The black pieces on this lamp were pretty good, but there were some white paint splatters on the atom-shaped part, so I used some fine steel wool to remove the white, and that led to the paint looking dull. So I decided to go ahead and prime then paint the base, atom, and brass tube so they'd all look fresh. I've found that satin finish is better than gloss for these projects, but use your judgment. And spray lightly with several coats. And wear a protective mask so you don't breath the fumes!
When you have all of the parts ready to go back together, walk through the reassembly steps. Lay the pieces out in the order they fit onto the lamp, and refer back to your photos to be sure you have the parts laid out correctly. Don't get too frustrated if you get the lamp mostly together then realize you left out a part or two, this has happened to me many times. Just back up, and start again. I find that once you have the new cord fed through the pieces that screw together, leave enough slack to screw the wires onto the socket, tighten the screws, and then reverse the steps you took to take the lamp apart. For more detail on rewiring, check out some rewiring how-to sites such as this one.
Before you get too far with your reassembly, after you've attached the new cord, put a bulb in the socket and plug in the new cord to be sure the bulb lights up. If it does, finish putting things together. If it doesn't light up, it's either a bad bulb, or you may not have the wires screwed down tightly.
Once you have the lamp nearly all together, make sure you don't have any leftover parts. The last thing to do is pull gently on the cord to make sure it has a little tension on the socket, so that nothing is wobbly. Tighten the brass nut on the base. Check the lamp again to make sure it works. Then you can either re-glue the old base felt or cork, or replace it with some new material of your choice. It's a good idea to do this so the base will be soft and won't scratch the surface the lamp will sit on. I've used felt and hot glue, but the adhesive-backed cork shelf liner is really easy to use. Cut out a piece larger than your base, stick it on, and cut around the perimeter to trim the excess.
After you fix a lamp yourself, you'll be more confident to try saving other old lamps, then your biggest problem will be finding a place for each of your lamps.
I'm waiting for the paint to dry before reassembly, that will be a future blog post if I can remember how this thing goes back together ;-)