A category is confluent if for any span $B \leftarrow A \to C$, there exists a cospan $B \to D \leftarrow C$. Note that we do not require the resulting square to commute.
If the morphisms in a category represent (sequences of) “rewriting” operations, then confluence means that any two ways to rewrite the same thing can eventually be brought back together. This is a good property of rewriting in systems such as the lambda calculus (the Church-Rosser theorem), and as such it is a property one might expect for the hom-categories in a 2-categorical model of lambda calculus.
Another good property one might want to assume is termination, i.e. the lack of infinite chains of nonidentity arrows.
John Baez, 2-categories of computation.
Barnaby P. Hilken, Towards a proof theory of rewriting: the simply-typed 2λ-calculus, Theor. Comp. Sci. 170 (1996), 407-444.