The name may be derived from the old Anglo-Norman word for "crab" and that it began as a nickname for a club-footed illegitimate son of an English princess by a Norman knight. A crab moves sideways and so the name could fit a child with club feet. Whether far-fetched or not, it is fact that at one stage the family crest was a crab (subsequently five feathers) and that the family motto is still "Devant si je puis" -("forward if I can"), which could have a double meaning as of course a crab can only go sideways.
One Richard Fitz Scrob (or Fitz Scrope), apparently a Norman knight, was granted lands by Edward the Confessor

Edward the Confessor

Edward the Confessor also known as St. Edward the Confessor , son of Æthelred the Unready and Emma of Normandy, was one of the last Anglo-Saxon kings of England and is usually regarded as the last king of the House of Wessex, ruling from 1042 to 1066....

 before the Norman Conquest, in Herefordshire, Worcestershire and Shropshire as recorded in the Domesday Book

Domesday Book

Domesday Book , now held at The National Archives, Kew, Richmond upon Thames in South West London, is the record of the great survey of much of England and parts of Wales completed in 1086...

.