There are a couple other instructables on how to repair the DC connector and how to repair the cord near the power supply.
My problem was a little different than either of theirs. I had repaired the connector in the past -- the cord frayed and shorted right near the DC connector (the part that goes into the computer) so I spliced it and tried making a better strain relief. That lasted for a couple years but it broke again so I figured I'd try and replace the connector entirely.
Step 1: Disassemble the Old Connector
Since Apple can change their minds anytime, I recommend that everyone disassemble their existing connector to decipher how the power supply connects. Mine is a "yo-yo" or "saucer" supply #M7332 that came with the Indigo "Clamshell" iBooks.
I used hot-glue as a strain relief in my last repair so I used a heat gun to get the glue off. I don't remember how the connector came apart originally, but the gist was to cut off the plastic shell then pry apart the metal shell. These are the pieces I came up with.
Step 2: Selecting the Replacement Connector
The connector is clearly a 1/8" stereo headphone plug with a custom metal shell around it. I bought a replacement plug but it didn't look like it would be easy to fit it inside the metal shell.
I decided to hedge a bet and just install the replacement jack without the original metal outer shell.
Although the outer shell was connected to the middle-ring and to the wire shield, I guessed that the shell itself was not connected for supplying power. I would have to take apart the iBook to find out for sure, but for the time being I just assumed it wasn't connected. If the resulting repair failed to get the iBook powered, then I could assume I was mistaken.
Step 3: Prepare the Wire
Now that the plan was established, I started wiring up the new connector. The first step was to cut the cord and prepare the cut end to be soldered. Check out the other pictures for details on how to strip shielded wire.
First, remove about a half-inch of outer insulation without damaging the wire braid. I find that if I use a sharp blade with light pressure against the wire, I can gently roll it along the blade and it will cut nicely. If it isn't obvious, it's easy to cut your thumb, so take your time and be careful.
Next, to separate the braided insulation, I use a pick to comb out the wires. Some electronic shops sell them as dental picks -- it's a handy tool to have. Once the wires are fanned out, I pull them to one side and twist them together.
Usually I'd leave a long tail on the twisted braid, but since I'd be connecting it "backward" (the braid connects to the inner-ring and the center-conductor connects to the shell/ground) I cut it short and tinned the ends.
Step 4: Solder on the New Connector
Next is to solder on the new connector. First, heat the tabs to connect with the soldering iron and tin on a blob of solder to each. For the connector shield, put a blob on the strain-relief bar. I found it easier to kink the center conductor and solder it first to the strain relief bar. Once I did that, the short stub of braid was nearly in place to connect to the middle-ring connector.
Step 5: The Moment of Truth
Check your work at this point and make sure nothing is shorted out -- or will short out when you assemble the connector housing. (Hopefully you didn't skip the part about putting the shell on the wire before soldering and can now assemble it.) On my connector there were bendable strain-relief tabs. Bend them gently over the insulation -- unlike 99% of times wiring a headphone jack, this time you don't want the braided shield to connect to the strain relief or it will short out the power supply.
Once everything was done, it was time to try plugging it in. As indicated by the orange glow, it appears to have worked!