Took the name of his step-father Henry Barham by Act of Parliament, according to Henry’s Will 631 FILE - Private Act, 22 George II, c. 14 - ref. HL/PO/PB/1/1748/22G2n31 - date: 1748
[from Scope and Content] An Act to enable Joseph Foster Barham Esquire, and his Issue, to take and use the Surname of Barham, pursuant to the Will of Henry Barham Esquire, deceased.606
Note that http://www.sharman.info/fam-shar/fam00316.htm inserts another generation here, which is not explicitly mentioned in http://www.archiveshub.ac.uk/news/0212cla.html561, 631