Is there a name for $(X,S)$ where $X$ is a set and $S\subseteq X$ and a morphism $(X,S)\overset{\alpha}{\longrightarrow}(X^\prime,S^\prime)$ is a function $\alpha:X\rightarrow X^\prime$ such that $x\in S\implies \alpha(x)\in S^\prime$?
I find this structure interesting since a lot of mathematical structures can be expressed this way.
If the sets have a topology, it's a pair of space. So you could call your sets is a pair of sets. The category you describe is sometimes described as the relative category of sets. I know no name for it, I would call it $\texttt{RelSet}$.
If the subset is always a single point (i.e. a singleton), it's called a pointed set and the category is denoted by $\texttt{Set}^*$.