I suppose it's technically possible, but it wouldn't be terribly convenient. You'd lose a lot of the benefits of the DS's compact design. There's also the usual peripheral dilemma, of course, of no games being developed for it because not everyone has it. I really don't think analog sticks are the best way to handle analog control on a handheld, anyway. I'm not sure what the best way is, but we need a new paradigm. We need something compact, resilient, and just as effective.
The touch screen doesn't work primarily for two reasons. First of all, it's not in a good position for your thumbs. Second, it has none of the tactile feedback that an analog stick has.
The PSP's analog control gets criticized a lot, but I haven't actually used it, so I don't know what lessons to take away from it.
My favorite idea so far is something that resembles the NES Max D-pad, but is self-centering and, obviously, an analog control instead of a D-pad. I just can't figure out exactly how to make it work. My second favorite idea, and one that could have worked on the DS, is similar to some PC game pads I've seen that have a stick you can screw into the center of the D-pad to make a sort of digital joystick. Make it a pressure sensitive D-pad and have the stylus double as the stick, and you've got something like an analog stick that would have worked well for the DS. It would certainly be compact and portable (you have to carry around a stylus with the DS already), but it wouldn't be quite as good as a real analog stick.