Which problems are hard for P^NP?
Krentel gave two problems complete for $\Delta_2^P$ (see Theorem 3.4):

Input: Boolean formula $\phi(x_1,\dots,x_n)$.
Question: Is $x_n = 1$ in the lexicographically largest satisfying assignment of $\phi$?
Input: Weighted graph $G$, integer $k$.
Question: Is the length of the shortest TSP tour in $G$ divisible by $k$?

Krentel also states that the only previous example of a $\Delta_2^P$-complete problem up to 1988 was Uniquely Optimal TSP, given by Papadimitriou in 1984.
As sdcvvc pointed out, see also the MO discussion at https://mathoverflow.net/questions/2218/characterize-pnp for a nice discussion by Ryan Williams of a machine model that captures $\Delta_2^P$.

Mark W. Krentel, The Complexity of Optimization Problems, JCSS 36 490–509, 1988. doi:10.1016/0022-0000(88)90039-6