So on the one hand I have a very strong cultural bias against undergraduate research programs. I don't think trying to emphasize originality is a good idea. I think it would be much better to give people problems to work on that have already been solved and so you know lead to good and interesting mathematics. By forcing people to work on "new" questions I think you are often forcing them to work on bad math.
On the other hand, just because I think it would be better for people to do other sorts of programs, REU-style programs are what exist and they seem to work reasonably well for a lot of people. Furthermore, they're certainly valuable as an alternative to classroom learning. Real math research is not like what happens at most REUs, but it's also not like what happens in a classroom, so doing an REU is still going to help you get closer to understanding the scope of what a graduate student does.
So yes it's certainly possible and somewhat valuable for undergraduates to try to do "research," but you shouldn't expect that research to be the same sort of research that mathematicians are really doing.
