The yoga of categorification has gained a lot of popularity in recent years, and some techniques for it have made a lot of progress. It's well-understood that, for example, a ring is probably categorified by a monoidal abelian (or triangulated) category.

But I get a little more confused another step up. If I have a braided tensor category, what sort of 2-category should I expect to categorify it?

Edit: I realized this wasn't the right question to ask; what I really wanted to know is What structure on a monoidal category would make its 2-category of module categories monoidal and braided?