Of course, my favorite example is the $2$-category of $2$-tangles (defined below) is a categorification of the category of tangles. The category of tangles is a monoidal category with objects that correspond to the non-negative integers, morphisms are generated by $|$, $\cup$, $\cap$, $X$ and $\bar{X}$. In this $1$-category, the Reidemeister moves (and zig-zag and $\psi$-move) are identities.
In the $2$-category of $2$-tangles, the $2$-morphisms are generated by ${ } \leftrightarrow O$ (birth or death), $| \ |\leftrightarrow \stackrel{\cup}{\cap}$ (saddle), and the aforementioned five Reidemeister moves (I, II, III, zig-zag, and $\psi$). These are subject to the full set of (35 or so) movie moves. The $2$-category of $2$-tangles is a braided monoidal $2$-category with duals. In fact, it is the free braided monoidal $2$-category with duals on one self-dual object generator (Baez and Langford).