I am pretty sure this problem has a name, or it can be reworded so it does. We are given a set $X$ and a family $\mathcal F$ of subsets of $X$, the problem is to find a subset $B$ such that $B\cap A\neq \emptyset$ for all $A\in \mathcal F$.
Does this problem have a name? Finding a subset intersecting all subsets of a family
co.combinatorics