For the most part, you get new wiring, connectors, a good soldering iron or gun, good
electrical solder, and a reasonable work area.
Lay out the old parts where you can see them, and cut your new wire a few inches longer than the existing wire. OR ~ cut the wires to your own needs. I have found that the average foot controller wire is about a foot shorter than I like it. So when I rewire one I add a foot.
Then carefully take each piece apart and draw yourself a schematic of how it's wired. This is especially important on the junction blocks and Singer motors. Some are wired simple, some are complicated. Even the simple ones can be easily miss-wired. (Don't ask me how I know this.)
Many times you'll find the connection ends are twisted and soldered into a ring then screwed down. You can do that, or solder in ring connectors. I prefer to use ring connectors when I can. First I crimp the bare connector, then I solder it. Probably over kill but it works.
I usually start with the foot controller. Then go to the light and then to the motor and lastly the connector block. That's just me, it can be done in any order.
The one thing I will emphasize is get good electrical solder. I usually buy it from Radio Shack. ( Hopefully they still stock it.) It's very high quality, melts quickly and makes very good connections.
Another tip is before you twist your wire for connections, or to put in connectors, wipe your fingers with Denatured Alcohol. This takes the body oil or any other oils off your fingers and makes it much much easier to solder the parts.
Inside the motor, be careful how you do the soldering. Sometimes there is a connection, other times there is just two ends of wire soldered together.
Learn to tie The Underwriters knot. There is really good instructions here:
http://vssmb.blogspot.com/2011/12/ho...or-part-6.html
Mater of fact read and study this blog about rewiring motors:
http://vssmb.blogspot.com/search/label/rewiring
Hope this helps a little.
Joe