Let us say that a family $R$ of sets has the Finite Subcovering Property --- FSP --- if any subfamily of $R$ which covers the union $\cup B: B \in R$ has itself a finite subfamily which also covers. For example, take for $R$ the family of open balls in a compact metric space $M$. Clearly the family of finite intersections of open balls also has the FSP.
We say the Finite Intersection Principle --- FIP --- is the statement: If $R$ has the FSP then the family of all finite intersections of members of $R$ also has the FSP.
It is easily proved that FIP is a theorem in ZFC. The question is: Does ZF + FIP imply the Axiom of Choice?
I looked in Herrlich, "Axiom of Choice", and I did not see this, but I may have missed it. This must certainly be known: the question was raised by J. L. Kelley in Fund. Math. 37 (1950), p. 76.
Note that in the absence of the Axiom of Choice, we need to specify what "finite" should mean. For our purposes here, let's say a set is finite if it may be ordered so that every non-void subset has both a first element and a last element in the ordering. See Herrlich Section 4.1 for equivalent formulations.