With a Nintendo joystick, the most important thing is to clean it thoroughly. The ball pivot socket is a trap for crud buildup. The easiest thing to do is take it apart and let the base (which contains the pivot ball encased in a socket) soak overnight in a solvent (gasoline is an effective solvent, and cheap enough to use in parts-soaking quantity).
Use stuff like a small brush, Q-tips, pipe cleaners, etc., to scrub down in the socket after it has soaked long enough to loosen the crud buildup.
If you use an oily type solvent like gasoline, kerosene, WD-40, etc., once you have all of the gunk out of the socket, clean off all the oily residue from within the ball socket with something like carburetor or brake cleaner. Those can be used for the initial cleaning instead of letting it soak in a solvent bath too, but you'll probably end up using a lot, and they're relatively expensive.
If your joystick has never been cleaned, you'll notice a world of difference after cleaning. You technically don't even lubrication, because there is pretty much no load or sustained rapid motion, but for good measure, I use 3M dry silicone spray on my Super Punch-Out joystick, and have for about 10 years. Whatever you use, you want it to be light. Silicone spray or light oil is fine (such as 3-In-One oil). A bonus with 3-In-One is that it smells awesome (almost as good as Hoppes No. 9 gun cleaning solvent). If you use grease, you only want a very light layer of it. If you pack the socket full of grease like you would with wheel bearings, it will make the joystick feel sluggish.
By the way, Nintendo never mentioned 3-In-One oil that I know of, but Atari did. It is what they recommended for the bearings in Missile Command's trackball (and maybe for other games too):
But 3M dry silicone spray beats everything in my experience. It adheres tenaciously to everything, even after the liquid carrier dries (hence the "dry" in the name), and is slippery as hell. Because it dries, it won't attract dust like grease and oil do. I was using some a couple months ago and some overspray got on my wooden floor. My foot still slips on that spot every time I walk on it.