From the new details in your lastest post I found more information-
1880 Census- Brooklyn, Kings, N.Y.:
Alfred Searle, 39, carpenter, born England; Ann Searle, 38, Ireland; William Searle, 6, N.Y.; James Searle, 12, N.Y.; Alice Searle, 5, N.Y.; Alfred Searle, 3, N.Y.
IGI has submitted details of the family-
Alfred Searle m.(1867 Kings Co.) Anna. Children- James, Joseph (b.1868 Kings Co.), William (b.1874 Kings Co.), Alice, Alfred (b.1877 Kings Co.).
Most importantly is an extracted record- Elizabeth Searle born 29 Aug.1880 Brooklyn- parents Alfred Searle and Ann Boyd
Italian Gnealogy website lists a death for Alfred Searle 19 Apr.1913 Kings Co. (age 38) which may be the Alfred supposed to have been born 1877. There's also a death for an Alfredo G. Searle 29 June 1885 Kings Co. (age 43) but not sure if this is Alfred Searle who married Ann Boyd.