When would the intersection of the empty set mean anything?

The following question come from remarks from Velleman, How to Prove it (pp.82, ex. 14).

Let $F$ denote a family of sets and consider the case where $F = \emptyset$. Then the statement, $x \in \cap F$ is always true, regardless of $x$. Thus the intersection of an empty set is the universe of discourse.

Paraphrasing from Velleman, the universe of discourse means different things depending upon the context and sometimes no universe of discourse is used. Thus some mathematicians consider $\cap \emptyset$ to be meaningless.

Since only some mathematicians consider it to be meaningless, this suggests that some mathematicians believe it to have meaning.

In what situations may the intersection of an empty set have meaning?

In set theories that have a universal set $\mathcal U$, the empty intersection is the universal set: $$\bigcap_{x\in\emptyset} x = \mathcal U.$$ As one would expect, one has $$\mathcal U\cup S = \mathcal U\\ \mathcal U\cap S = S$$
for all sets $S$. The set theory most commonly studied today is ZF, which doesn't have a universal set, but other set theories have been proposed. One such is Quine's NF ("New Foundations"). If you're interested you can read Mark Holmes' monograph Elementary Set Theory with a Universal Set, which is about a better-behaved variation of NF known as NFU.