According to
Dictionnaire etymologique des noms de famille et premoms de France by
Albert Dauzat, the French surname Marat normally indicates one's ancestors came from a place in the Puy-de-Dome area of
France. However, in the case of the famous revolutionary Jean-Paul Marat, his name is a "Francization" of the Sardinian surname
Mara, as his father was from Sardinia. The revolutionary's family in turn claimed that
Mara was originally a surname from Spanish nobility, but Dauzat thinks a Jewish origin is more likely. But in either case he doesn't give an original meaning.