Most likely named after the cliffs of nearby Grymes Hill.
The name dates to 1817, when a town called Clifton, larger than today’s neighbourhood, was laid out along the waterfront. Forgotten New York suggests it was most likely named for its position east of the Grymes Hill cliffs. The station opened in 1860 as Vanderbilt’s Landing: Cornelius Vanderbilt had started a ferry from this waterfront to Manhattan as a young man.