There's a categorification of what it means to be an element of a set; indeed, from Goldblatt's Topoi: The Categorial Analysis of Logic, we have this
Definition: In any category $\mathbb{C}$ with terminal object $1_{\mathbb{C}}$, an element of a $\mathbb{C}$-object $a$ is a $\mathbb{C}$-arrow $1_{\mathbb{C}}\stackrel{x}{\to}a$.
The idea of categorification is, that of replacing sets with categories, elements with objects, relations between elements with morphisms between objects, etc., so perhaps the above isn't quite a categorification of an element.
There's the category of rings and ring homomorphisms.
So are there categorifications of the notions of prime or irreducible elements (of a ring, say)?
An object $X$ of a category is called indecomposable if $X$ not initial and for every coproduct decomposition $X = A \oplus B$ we have that either $A$ or $B$ is initial. Similarly, $X$ is called simple (or irreducible) if $X$ has exactly two quotients. Both notions are very common in representation theory. In Krull-Schmidt categories we have categorified prime factorizations.