Sure. A $15 controller to an OEM (made by Nintendo) is probably the biggest jump in quality that a regular person will notice. The aftermarket controllers often feel cheaply made, skimping on things like replacing the rubberized stick caps with hard plastic. I've seen controllers where the plug was just a couple millimeters too large, causing it to be stuck in the console port. The stick boxes often sometimes feel coarse and wear out quicker than they should.
On the upper end of the spectrum, OEM controllers from Nintendo are generally considered high quality, but in a precise game like Melee there are a couple things that can cause hangups.
The analog stick might not be properly calibrated. The controller reports the position of the analog stick as its x and y coordinates from -1.0 to +1.0 in both axes. Pushing the analog stick all the way to the edge should result in a 1.0, but some controllers instead report a 0.9875. Completely unnoticable for 99.9% of players, but at the highest level it can make a difference. Sometimes fixed by shaving more plastic off of the controller shell, but custom PCBs let you calibrate the stick manually.
"Snap back" is a more common issue. Tilt the stick all the way to the edge, then let go. Ideally, the stick would return to the center, but oftentimes the momentum carries it past the center into the opposite axis, resulting in a small input in the opposite direction. For some techniques this can have bad consequences, such as Falco needing a neutral input to shoot his gun, but a small input in the opposite direction will turn him around and shoot the gun the opposite way. Common fix for this is soldering a small cheap capacitor to the output of the stickbox's horizontal potentiometer, very subtly smoothing out the directional output.
Fancier mods include things like reducing the travel distance of the gamecube controller's large shoulder buttons to allow for quicker inputs or replacing the shoulder Z button with a clickier button.
This youtube video goes over a popular custom PCB that people have started to use in place of OEM, and some of the reasons why it is desirable: https://www.youtube.com/watch?v=RGyQbXGr4Po
If you just want a decent controller, finding a normal Nintendo-made gamecube controller at a used games store or ebay isn't that hard yet. Odds are it'll still be in decent shape.
If you want to buy a custom one, I don't know if I have any great answers for you. It's a pretty grassroots community, so depending where you live, your best option might be to scour facebook/twitter/discord to find your local Melee players and ask for the guy that makes/sells controllers.