Categorification of algebra structures This might be a bit of a soft question.
Take a $\mathbb{C}$-linear category. Form the complex vector space spanned by its objecs modulo exact sequences. This construction is, as far as I know, the reason why $\mathbb{C}$-linear categories are thought of as categorified vector spaces.
Now put a monoidal structure on the category. This gives a multiplication on the vector space from before, making it into the Grothendieck algebra of the category. So monoidal structures on linear categories are categorifications of associative algebras. The property of the algebra being associative corresponds to structure of the associator on the category, as we would expect in a categorification.
As a further warmup, a commutative algebra categorifies to a braided monoidal structure. Again the property of commutativity is categorified to the braided structure in the category.
Now, imagine an involutive algebra. This is the strange bit. An involution is structure on the algebra. But the categorification is a property on the category, the existence of duals! (The dual of an object gives its involution in the algebra, and two duals to the same object are always isomorphic, so it's well-defined.)
What's going on? Why is the usual order of stuff, structure and property reversed? Are there other examples of this phenomenon?
Remark: I'm lying a bit, the property of the involution to square to the identity categorifies to a pivotal structure, which is a monoidal natural transformation from the identity to the double dual.
 A: Here's a simpler example: in ring theory, picking a basis of your ring (as a module over the ground commutative ring $k$) is extra structure. But in category theory there is sometimes a "distinguished basis" (e.g. the simple objects in an abelian category), which will pass to just a basis after decategorifying. 
For example, Hecke algebras have a famous basis called the Kazhdan-Lusztig basis which is used to define Kazhdan-Lusztig polynomials, which are important in representation theory. It turns out that this basis can be thought of as coming from a categorification of Hecke algebras using what are called Soergel bimodules. 
So category theory sometimes furnishes "property-like structure" (stuff that's not preserved by functors but is uniquely determined by categorical considerations) in a way that's invisible and just looks like some random extra structure when you decategorify. This is arguably a major reason to care about categorification: to see how the category theory suggests more structure than is visible at the decategorified level. 
