The Main Purpose: Keeping Your Clipper Attached
On many nail clippers, the small hole is designed as an attachment point.
You can thread a small split ring, keyring, cord, or lanyard through it. That allows you to attach the clipper to a travel pouch, grooming kit, hook, or another convenient location.
This solves a very ordinary problem: nail clippers are small, lightweight, and incredibly easy to lose.
Instead of disappearing into the bottom of a bathroom drawer, the clipper can have a permanent home.
Similar explanations of the feature identify attachment and storage as its primary practical use.