The connection that goes from the car to the blower motor in the glove-box has an issue to where if it's not twisted in the right position the blower will not turn on. I constantly have to readjust it to make an electrical connection so the blower starts. It's getting really annoying and I'm running out of ways to twist the wire. The glove box itself has been removed for easy access to the wires.
Has anyone else had this issue and would I need to have the blower rewired?
Is it possible the plug pins are corroded and not making good contact? I've found a little piece of sandpaper wrapped around a tiny Phillips head screwdriver, piece of wire, or nail works well for cleaning the female ends. Of course, make sure the wires don't have power while you do this. Otherwise, I am not sure. It could be a broke wire or solder connection at the plugs.
Scott has pretty much told you what the issue probably is. Definitely clean the male and female connections, but also pry open the bi-polar phenolic plug housing and make sure that none of the wires have broken from their soldered connection in the metal pin connector.
As far as your actual blower motor goes, it's not something one can rewire. Inside of the motor are an armature, a commutator, carbon piles, field magnets, and bearings with shims on each end of the armature shaft. The armature is comprised of solid copper wire would over and over and coated in varnish. It's very unusual for the armature to be bad, and to test the armature one has to have what is called a "Growler". Back in the days when people actually rebuilt alternators, starter, and various other motors, test equipment like growlers was commonplace. Not so much anymore. With enough use, the carbon piles will wear down and cause intermittent operation. When this happens one can often strike the blower housing lightly with their hand and the motor will start. If one knows the dimensions of the carbon piles they can be procured at little expense. Replacing them does involve taking apart the blower motor, and while inside the bearings should be cleans and lubricated, and the commutator should be visually checked for excessive wear or carbon traces (a sign of excessive arcing). The commutator is what the carbon piles rub against as the motor turns.
If after cleaning the electrical connecting plug and checking the soldered connections in the plug the issue is still present, then I'd either source another blower motor or seek out a shop that specializes in repairing electric DC motors......if there are any left in existence.