The claim relies on an excerpt from Meteorology, written by Aristotle, that a rock the size of a wagon fell to earth in the second year after the 78th Olympiad, and that a philosopher named Anaxagoras predicted it. This caught my attention as much because I use Anaxagoras as a character in The Ionia Sanction! Anaxagoras was a pre-Socratic philosopher. Though only just pre-Socratic; young Socrates was aged between 2 and 4 years old when the meteor hit. Anaxagoras was also well known among the ancient Greeks for his wild idea that all matter was composed of infinitesimally small particles.
The suggestion that Anaxagoras could have predicted a meteor fall is obviously rubbish. But Aristotle mentions in passing that a comet was visible in the sky when the rock fell. As it happens this is in the window for when a Halley's Comet flyby would be expected.
The best you can say is it might have been Halley's comet. If so, it's the earliest known sighting.