Earl Close is off Henry Street and continues the memory of Earl Street which was built c. 1815 and marked on Mason's map (IHTA xii, 14). It was the next street north, parallel to Henry Street. It is unclear which earl is referred to. By this time, the title of Earl of Donegall had been elevated to Marquess, but the Marquess' heir held the title Earl of Belfast, so this is a possibility.
Royal Mail Postcode Finder gives Earl Close as an address within Henry Street.