As noted elsewhere, this seems to have been also known as Brooke or Brooke's Court, based on the fact that Boyer Brooks (or Bowyer Brooke) lived here both in 1785 and 1791; he probably owned the whole property and built rental properties on it. Note that he was a boat builder, and the corresponding high proportion of maritime-related jobs in the 1791 listings. White did not cover this alley.
1785 (MacPherson) listings:
(1785) #2, Taylor, John
(1785) #3, McDonald, Widow
(1785) #4, Carr, Widow
(1785) #5, Brooks, Boyer
1791 listings:
(1791) #2 Brooke Court, Murdock, James, labourer
(1791) #3 Brooke Court, Crawford, Samuel, sea captain
(1791) #4 Brooke Court, O'Donnell, Rose, widow
(1791) #5 Brooke Court, Carlisle, Abraham, silversmith
(1791) #6 Brooke Court, Nailor, Lane, ship carpenter
(1791) #7 Brooke Court Brooke, Bowyer, boat builder, also at #109 N. Water St.