Just a week or two ago, I took apart a N64 controller. The problem it was having was that the stick pivots around a sphere with grooves on either side, those grooves containing small plastic cylinders from the casing. The problem was that the tops of the grooves had been worn out, causing the spring to push it all further into the controller, unlinking the x/y gearing, except at extreme tilts.
Repairing one properly would be as simple as replacing a single plastic part. (and cleaning out the dist from the old one). Quite durable, except for that one part that was under constant force and friction for years of abuse from three people frequently playing SSB and to a lesser extent some other games... (I didn't have a replacement, so I put in something to temporarily restore partial functionality, though I think it wore out already, so that my younger sister could enjoy LoZ:OoT and actually be able to move faster than a walk. Rather disappointing that the gamecube/wii doesn't have backwards compatibility for N64 games, though I appreciate the capabilities and simplicity that they gained by not doing so)