Under popular derandomization assumptions, the following problems are in $NP\cap coNP$:
• Group Membership (e.g., given invertible matrices $A$ and $B_1,...,B_k$, is $A$ in the group generated by $B_1,...,B_k$?)
(More precisely, these problems are known to be in $NP\cap coAM$. $coAM$ is a "close cousin" of $coNP$, and equals the latter under derandomization hypotheses: see this paper by Klivans and van Melkebeek.)
Besides factoring, there are various other number-theoretic problems in $NP\cap coNP$, such as decision versions of Discrete Logarithm (both in $Z_p^*$ and in elliptic curve groups).
If you're willing to allow promise problems (i.e., the algorithm only has to output a correct answer if the input satisfies some property), then there are lots of natural examples of $NP\cap coNP$ problems. A trivial example is, "given two Boolean formulas F and G, and promised that exactly one of them is satisfiable, decide which." A nontrivial example is the Approximate Shortest Vector problem, mentioned previously by Niel. What's rarer are interesting $NP\cap coNP$ problems that don't have a promise (or where the promise is easy to check).