This note represents a rare misstep by Dijkstra: the pigeonhole principle actually is a special and important principle worthy of a special name of its own. For example:
When you formulate the pigeonhole principle in propositional logic, its resolution proofs are exponentially large. Since modern SAT solvers are basically very fancy propositional resolution provers, this gives you a nice way to find hard instances for them. It also makes the question of when propositional proof systems can be more succinct than one another is also a fundamental question in complexity theory.
It also arises in geometry: a compactness for metric spaces is essentially the statement that the pigeonhole principle applies to the space.
I don't have a unified perspective of these two facts, but either one of them is really striking. As a result I'm okay with giving the pigeonhole principle its own natural language name.