From Old French borgne "one-eyed, blind", a nickname for someone with only one eye, or who had other problems with their vision, such as a squint or cross-eyes.
Shortened form of the Dutch surname van den Borne, derived from Middle Dutch borne "well, spring, source". A habitational name for someone from Born in the province of Limburg (Netherlands) or from a place associated with the watercourse of the Borre river in French Flanders.