I bought a spare stick on eBay and a set of tools the tri-wing heads from Amazon. Together they were about $15. My system's well past warranty and I heard it would be many times that plus turnaround time if I sent it to Nintendo so I took a chance and tried it myself.
Once I had part and tools it took about 20 minutes to take it apart, swap out the stick, and put it back together. Fortunately nothing needed soldering or anything like that.
It was the first hardware problem I've had with Nintendo that I didn't cause. I tried many other things before this though. I made sure the gamepad was level when I turned it on, cleaned off the thumbstick dome as much as I could, blew compressed air in it in case it was just some crud that got in there, but nothing else worked. The tools arrived first and I tried just opening it up but there was nothing obviously wrong with the stick assembly so I blew compressed air in it and put it back together. At first it was OK but then it started up again. So when the part came in I put it in and so far it's worked well.