I'm looking for a small research problem an undergraduate would be capable of after taking just an abstract algebra course, introductory algebraic geometry (at level of Miles Reid's book and Ideals, Varieties & Algorithms), and a course in number theory. Is there a website that would have a decent listing, or possibly a book one can recommend that may have small open research problems?

Your question is missing a crucial word in the first sentence (capable of ...). Is the missing word "understanding" or "solving"? Anyway, here is a problem: Find the maximum number of points of a curve of genus $g$ over $\mathbb{F}_q$, for some values of $g,q$ for which this number is not known (check for values at http://www.manypoints.org/ ) 

