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.
Best Answer
Here is one set of data that will be sufficient. To get the monoidal structure you don't actually need a (monoidal) functor $C \to C \boxtimes C$. It is sufficient to have a bimodule category M from C to $C \boxtimes C$. You will also need a counit $C \to Vect$, and these will need to give C the structure of a (weak) comonoid in the 3-category of tensor categories, bimodule categories, intertwining functors, and natural transformations.
To compute what the induced tensor product does to two given module categories you will have to "compose" the naive tensor product with this bimodule category. This can be computed by an appropriate (homotopy) colimit of categories. It is basically a larger version of a coequalizer diagram. This is right at the category number where you will start to see interesting phenomena from the "homotopy" aspect of this colimit, which I think explains the funny behavior you're noticing with regards to bases.
Finally, you may get a braiding by having an appropriate isomorphism of bimodule categories,
$ M \circ \tau \Rightarrow M$
which satisfies the obvious braiding axioms. Here $\tau$ is the usual "flip" bimodule.