Hereditarily-finite sets are becoming increasingly more popular in computer science research.

Why? Because some ill-advised first year maths lecturer told you that the element relation was the foundation of mathematics, maybe?

"object" is a hereditarily-finite set plus some structure on the set and a "morphism" would be a structure-preserving function.

If you're really interested in heredity, so the "structure" is the element relation, this is a well-founded coalgebra for the covariant powerset functor. Coalgebras for the powerset functor were first studied by Gerhard Osius in JPAA in 1974, although he considered recursion rather than induction. Well founded coalgebras for general functors (but with some emphasis on the powerset) are defined in Section 6.3 of my book "Practical Foundations of Mathematics" (Cambridge University Press, 1999). The exercises for that chapter show how various ideas with recursive programs may be expressed in these terms. In particular, unary recursion (with at most one recursive call at each level) is reduced to tail recursion (equivalent to while programs) together with an accumulator monoid. Paul