Gresham Street was known as Hudson’s Entry and Hudson’s Lane on 1791 map. It had also been called Smithfield Street. "Resolved that the Council be recommended to name the new street from Richie's Place to Smithfield Market Smithfield Street". (17th December 1879). "Resolved that Mr Henry Hanna be informed that as the owners of property object to the name Smithfield Street for the new street from Ritchie's Place to Smithfield the Committee will name it Gresham Street if agreeable to them. (21st January 1880).
In the minutes of the Town Improvement Committee 2 February 1880, there is the following decision: “…at the request of the owners of property adjacent, the name of the new street from North Street to Smithfield has been named Gresham Street”. It is reported in this way in the Belfast Morning News on Tuesday 03 February 1880:
"They have named four new streets near Hillview Street, Bourton Street, Lebanon Street, Jordan Street, and Druse Street, and a new street off Rowan Street, Craigavad Street, and, at the request of owners of property adjacent, the name of the new street from North Street to Smithfield has been named Gresham Street".
Surmise: Sir Thomas Gresham https://www.layersoflondon.org/map/records/the-royal-exchange-9e334a14-18f2-40c8-932b-8e4c18575d8f