In ZFC each set $S$ has a well-founded membership tree building $S$ up from the empty set $\emptyset$. You could attach the membership tree for any given set $A$ on each of the bottom nodes for the tree of $S$ and collapse to get a set. I could call that new set $S(A)$ and say it is built from $A$ the way $S$ was built from $\emptyset$.
Is there some standard terminology for this process, and standard notation for this set $S(A)$?
I don't think there's a standard notation, but you can define one for yourself by well-founded recursion over all the sets. Along the way you get to specify exactly what you mean by "built from $A$ the way $S$ was built from $\varnothing$" -- for example either
$$ S * A = \begin{cases}A & S=\varnothing \\ \{ s*A \mid s\in S \} & S\ne \varnothing \end{cases} $$
or
$$ S \circledast A = A \cup \{ s\circledast A \mid s \in S \} $$