In addition to being a metonymic occupational name for a huntsman, the surname Chase could also be a nickname for an exceptionally skilled huntsman, and it is derived from Middle English Chase meaning "hunt" (Old French chasse, from chasser meaning "to hunt", Latin captere).
As an English surname, Chase is a metonymic occupational name for a huntsman, or rather a nickname for an exceptionally skilled huntsman, from Middle English chase "hunt" (Old French chasse, from chasser "to hunt", Latin captare).