John Cooke, "of New England, mariner" left a WILL in St Michael's Parish, Barbados, dated 3 Apr 1694, proved 19 Apr 1694, in RB6.2, p. 117.
"John COOKE of New England, mariner. St Michaels Parish. Bro Edward COOK living in West Grimstead in Sussex [England] - my estate now in Jersey; Sir Edward ANDREWS the Gov. Of Virginia; Abraham TOWDISEND [Townsend]. signed John COOKE. Wit: Israel WOSS, Arndoll JEFFORD, John TRANCK...."
If he had an Estate in New Jersey, then surely there would be some record of it. AND, therefore, a genealogy of THESE Cookes might be out there.