Magnet wire is thin, solid copper wire with an insulative coating of enamel, varnish or polymer. You can make your own magnet wire by drawing copper wire to a desired thickness and insulating it.
Drawing Wire
You can reduce the diameter of a length of wire by pulling it through a draw plate -- a sheet of metal with holes drilled in it. Hold the plate in a vice while you pull the wire through successively smaller holes until you have the diameter you want.
Wire Coating
To turn plain copper wire into magnet wire, you will need to give it a coating of insulating paint or varnish. The coating must be free of imperfections -- as any gaps could cause a short circuit -- and must also be flexible so it doesn't crack when you wind the wire.
Considerations
Making your own magnet wire is labor-intensive. Since magnet wire is cheap, it's not usually worth the effort of making your own unless you only want small amounts.