I recent discovered on Reddit (aptly described by a friend as "surfing 4Chan with a condom") that you can improve the New 3DS's crappy C-stick to some degree by popping off the rubber nub and replacing it with the analog thumbstick from a PSP-1000. As an outspoken critic of the C-stick, and being exceedingly bored, I decided to give it a shot. Here now are the results:
And here's the verdict: it's not tough to install, it stays put once it's pushed into place, and it really is an improvement over the standard C-stick. On the other hand, it doesn't work the way you might expect. Rather than tilting, the stick acts like a very small touchpad, reacting to your thumb with only a small amount of pressure. It's responsive, but not as tactile as the circle pad on the left. So it's not an ideal solution, but until Nintendo pulls its head out and does things right with the NX, it's as good as you're going to get.
Some notes: I'm not responsible for any damage you might do to your 3DS if you decide to try this yourself, but in way of assurance, I will say that such damage is fairly unlikely. Second thing... the C-stick cover is held in place with two rubber loops, which will snap when you tear it out of the system. The cover can be put back in place if you snip off the loops, but it won't be as secure without the loops anchoring it in place. Finally, the analog thumbstick has to be from the PSP-1000... sticks from later models of the system won't fit. They're not hard to find, though... do an eBay search and you should be able to buy one for maybe a couple of bucks.
Special thanks to Reddit for the information, and to the Talking Time forum for alerting me to its existence.