A braided monoidal 2-category is a 2-category equipped with a tensor product 2-functor which satisfies the first in a hierarchy of conditions for being commutative up to equivalence: in the language of k-tuply monoidal n-categories a braided monoidal 2-category is a doubly monoidal 2-category.
As described there, this may be identified with a tetracategory with a single object and a single 1-morphism.
Ben Webster: I would very much like to know: what structure on a (triangulated/dg-/stable infinity/whatever you like) monoidal category would make its 2-category of module categories? (give that phrase any sensible construal you like) is braided monoidal.
If one decategorifies this question, one gets the question “what structure on a ring makes its category of representations braided monoidal” and the answer to this question is well-known: a quasi-triangular? quasi-Hopf? structure.
I asked a MathOverflow question on the same topic. No interesting answers yet.