Given a collection of sets $F$, a set which intersects all sets in the $F$ in at least one element is called a blocking set (or hitting set). The blocking number $τ(F)$ of a family $F$ is the minimum number of elements in a blocking set of $F$.
Let a set which intersects all sets in the $F$ in at least two element is called a 2-blocking set; and The 2-blocking number $τ_2(F)$ of a family $F$ is the minimum number of elements in a 2-blocking set of $F$.
Has this variation of blocking set been Studied Before?
Yes. They are called 2-blocking set or double blocking set. More general, there is the term $n$-blocking set or $n$-fold blocking set.
To give you an example where double blocking sets have been studied: It is an open problem to determine the minimum possible size of double blocking in the projective plane $\operatorname{PG}(2,p)$ of prime order $p$. It was conjectured that a minimum double blocking set is given by a line triangle, which has size $3p$. However, in $\operatorname{PG(2,13)}$ there is a double blocking set of size $38$, beating the triangle by $1$.