Module categories over a braided tensor category C form a tensor 2-category Mod(C). Its 2-center is a braided tensor 2-category. We describe the objects of this 2-center as braided C-module categories. Any such a module category gives rise to a representation of a type B Artin braid group. We also introduce a higher Morita theory for braided tensor categories, where a Morita context for pair of braided tensor categories C, D consists of their embeddings into a non-degenerate braided tensor category whose images centralize each other in the sense of Mueger. We relate higher Morita equivalences between C, D with braided 2-equivalences between the 2-centers of Mod(C), Mod(D).