What's this operation on an element of a set of a family called?

First, you should clarify the context of how did you encounter this notion and why you think this notion is important to you. Not every question contain these factors, but this is important in your case since few people are familiar with the notions in your question. Clarifying the context would be helpful to get better feedback.

I also agree with the comment by @Ray that not every notion has a name. In fact, not everything must have a name: if we name something that is not important, we would forget it.

However, there is an algebraic characterization of the notion in your question that seems worth mentioning. Remember that the symmetric difference $$A\mathrel{\triangle}B := (A\setminus B)\cup(B\setminus A)$$ makes the set $\mathcal{P}(S)$ of all subsets of $X$ an abelian group. That is, $(\mathcal{P}(S),\triangle)$ is an abelian group. Then your operation corresponds to a pointwise addition (symmetric difference) of a singleton.

In the case of your example, let $S=\{1,2,3,4\}$, $F=\{\varnothing,\{1,2\}, \{3,4\},\{1,2,3,4\}\}\subseteq \mathcal{P}(S)$, and $a=\{2\}$. You can see that $F_2=\{x\mathrel{\triangle} a\mid a\in F\}$.