So, many of us know the answer to "what kind of structure on an algebra would make its category of representations braided monoidal": your algebra should be a quasi-triangular Hopf algebra (maybe if you're willing to weaken to quasi-Hopf algebra, maybe this is all of them?).
I'm interested in the categorified version of this statement; let's say I replace "algebra" with a monoidal category (I'm willing to put some kind of triangulated/dg/$A_\infty$/stable infinity structure on it, if you like), and "category of representations" with "2-category of module categories" (again, it's fine if you want to put one the structures above on these). What sort of structure should I look for in the algebra which would make the 2-category braided monoidal?
Let me give a little more background: Rouquier and Khovanov-Lauda has defined a monoidal category which categories the quantized universal enveloping algebra $U_q(g)$. It's an open question whether the category of module categories over this monoidal category is itself braided monoidal, and I'm not sure I even really know what structure I should be looking for on it.
However, one thing I know is that the monoidal structure (probably) does not come from just lifting the diagrams that define a Hopf algebra. In particular, I think I know how to take tensor products of irreducible representations, and the result I get is not the naive tensor products of the categories in any way that I understand it (it seems to really use the categorified $U_q(g)$-action in the definition of the underlying category).
To give people a flavor of what's going on, if I take an irreducible $U_q(g)$ representation, it has a canonical basis. It's tensor product also has a canonical basis, but it is not the tensor product of the canonical bases. So somehow the category of "$U_q(g)$ representations with a canonical basis" is a monoidal category of it's own which doesn't match the usual monoidal structure on "vector spaces with basis." I want to figure out the right setting for categorifying this properly.
Also, I started an n-lab page on this question, though there's not much to look at there right now.