He was the second son of Lieutenant-Colonel James (nick-named "Prime-Iron") Rochfort (d. 1652), a Cromwellian soldier, and his wife Thomasina Pigott, daughter of Sir Robert Pigott of Dysart Manor, County Laois, and widow of Argentine Hull of Leamcon, County Cork. Robert was born posthumously: his father, who had fatally wounded one Major Turner in a duel, was court-martialled and executed for murder a few months before Robert's birth. His mother made a third marriage to George Peyton of Streamstown, County Roscommon, who was her distant cousin through her mother Thomasina Peyton, second wife of Sir Robert Pigott.