I checked this. I have the official Mayflower Society Silver Book. In the official genealogy of the General Society of Mayflower Descendants William3 Hopkins (Giles2, Stehpen1) is listed on page 9:
"9 Jan 1660; Giles left land and cattle to son Stephen with condition that "after my decease my son Stephen Hopkins shall take ye care and oversight and maintain my son William Hopkins during his natural life in a comfortable and descent manner." Giles also left two acres to his wife and son William for their life use, to go to son Joshua upon their deaths; William was evidently unm. and d.s.p. after 5 March 1688/9; no proof has been found and it appears most unlikely that this man was identical with Wiliam Hopkins (1660-1718) of Shelter,NY, who m. Rebecca Havens."
Drat! The William who was son of Giles was clearly non compos mentis and therefore never married or had children much less ventured to New York.
Another theory I read claims he was son of Caleb2 but I looked this up as well. The book shows no descendants of Caleb. p 3 says that Caleb was dead by spring 1651 so that would be he could not have been father of William born five years after his death. It says he "bacome a seaman & dyed at Barbados," quoting Gov. Bradford between 6 Mar and 3 Apr 1651; d.s.p. in Barbados after 6 Jun 1644 and bef. 3 Apr 1651.
If I were you I'd be checking familysearch and Findmypast for records in England and see what you can find. Maybe there is a probate record naming a son William in New York.
