What structure on a monoidal category would make its  2-category of module categories monoidal and braided? 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.
 A: 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.
A: (I don't think I will say anything you don't already know, but I'll give it a shot anyways.)
Well, we get a notion of module category once we pick a symmetric monoidal 2-category in which to work.
If we pick a cartesian one like Cat, where monoids are just monoidal categories, then of course every 2-category of module categories is symmetric monoidal via the diagonal action.  So, I guess you're not interested in this case; I assume you'd prefer a more "linear" setting such as the 2-category of presentable categories.
For a general V, if C is a monoid in V, we should be able to make its 2-category of modules into a monoidal 2-category if we have a V-bialgebra structure on C in the obvious sense.  I say "obvious" because none of the diagrams I am thinking of have any non-invertible 2-morphisms.  Note that what we are really doing is looking for monoidal structures which on the underlying objects become the monoidal structure in V.  There are of course other possibilities such as "tensor over C".  (In fact I would have finished the first sentence by saying "the algebra should be an E3 ring spectrum".)
I don't really understand the R-matrix story well enough to see whether it should extend easily to this situation.  I would first want to understand, if V is say a presentable closed symmetric monoidal 1-category, what extra structure do I need on a Hopf algebra object A of V to make A-mod braided monoidal?  (E.g., the answer should be trivial when V = Set.)
A: In response to your first question, yes; a monoidal category structure on Rep(H) for an algebra H is equivalent to having a quasi-bialgebra structure on H.  And braiding gives you what's called a QTQBA.  If you also ask for duals, you get a quasi-Hopf algebra and QTQHA respectively.  You can recover the axioms for the associator Φ and the R matrix as endomorphisms of the functor to vector spaces, similar to Tannakian type constructions. Also the antipode S which satisfies rather strange looking identities. [edit: Earlier I misread the parenthetical part assuming it's already Rep(A) of some algebra, and then said almost the opposite =).]
Before talking about braiding, we'll want to ask the category have a tensor product structure on its 2 category of module categories, which will of course be extra structure.  I guess this will be equivalent to asking for a functor from \Delta: C to C x C satisfying various categorical versions of the bialgebra axioms.  There should be a co-associativity constraint, which should be coherent in some straightforward way, and there should be an isomorphism of functors categorifying the assertion that the coproduct of a bialgebra is an algebra morphism from A to A ⊗ A.  All of this together should endow your 2-category of modules with the structure of a monoidal 2-category.
You will then want a natural transformation of functors from \Delta\to \Delta^{op}, also satisfying various naturality properties.
How does that sound (as a start)?
A: I haven't spent as much time on this good question as I wanted to.  I also haven't spent  nearly as much time on the important topic that it comes from as I wanted to.  But here is a stab at it.
First, Khovanov and Lauda don't actually categorify the true $U = U_q(\mathfrak{g})$.  Lusztig doesn't actually find a canonical basis for it either.  Instead, they work with an atomized form $\dot{U}$ that, as far as I can tell, amounts to the representation category of $U$.  If I'm right about that, then the categorification gives you no hint as to how to decorate the Hopf category to make its 2-category of representations happy, because the question is tautological.  One thing that is going on is something that you identified as a hint of trouble in your question:  Given two irreps $V_\alpha$ and $V_\beta$, there are actually three basified combinations, using the tensor basis, the canonical basis, and the dual canonical basis.  I'm going to guess that they have categorifications with functors between them like this:
$$\mathcal{V}_\alpha \diamondsuit \mathcal{V}_\beta \leftarrow
\mathcal{V}_\alpha \otimes \mathcal{V}_\beta \to
\mathcal{V}_\alpha \heartsuit \mathcal{V}_\beta.$$
(Maybe it is not as simple as these arrows, but there is hopefully some sort of association.  Also I don't feel like using the symbol $\boxtimes$, but maybe I should.)
I think that the paper Khovanov-Lauda I has a true categorifigation of $U^+ = U_q(\mathfrak{g}^+)$ with the structure that you reject as the wrong answer.  Namely, their induction and restriction functors are functors
$$M:\mathcal{U}^+ \otimes \mathcal{U}^+ \to \mathcal{U}^+ \qquad \Delta:\mathcal{U}^+ \to \mathcal{U}^+ \otimes \mathcal{U}^+$$
So maybe $\mathcal{U}^+$ does act on all three of the above categorical products, and maybe $\mathcal{U}^-$ does too, but the actions don't fit together perfectly.  It's known that you can understand a representation of the decategorified $U$ as a certain kind of twisted Hopf module of $U^+$ (using the quantum double formula of Drinfeld), or equivalently as a twisted bimodule of $U^+$ and $U^-$.  Maybe the quantum double formula should be categorified in some way that does not lead to a categorification of $U$, since people haven't constructed one.
This suggestion is a bit like Chris' suggestion of a Hopfish categorification, but it would instead be a Hopf categorification that is double-ish.  Also it might only be braided-ish, since that comes from the quantum double.  Note that Khovanov and Lauda also didn't categorify the antipode in their paper, just the bialgebra structure.
A: Ben, I recently happened on the paper of Lyubashenko from 1999 (!), which discusses many of these topics, and even is motivated by the desire to categorify things like canonical basis etc.  It also contains references to works of Crane and Frenkel, and of Yetter, in this direction.  The paper is Lyubashenko, called
"Operations and isomorphisms in a triangulated Hopf category"
I hope you find this helpful.
A: Look at this great article:
http://arxiv.org/abs/math/0512165
