As a warmup, an $\mathbb{N}$-graded ring is a monoid object in the symmetric monoidal category of $\mathbb{N}$-graded abelian groups under the convolution tensor product, which you can think of as Day convolution from the usual addition on $\mathbb{N}$.

Similarly, this thing is a monoid object in the symmetric monoidal category of species in abelian groups (presheaves on the category $S$ of finite sets and bijections valued in abelian groups) under the convolution tensor product, which you can again think of as Day convolution from disjoint union on $S$. 

So I might be inclined to call such a thing an $S$-graded ring, and you can feel free to replace $S$ with your favorite name for $S$, maybe $\text{FinSet}^{\times}$ or something.