The first newspaper mention of any of the "Ashley streets" Ashley Avenue/Drive/Mews was Ashley Avenue in the Belfast News-Letter on 16 Aug 1860. Two years later there was an advert for an auction for Ashley House in the Belfast Morning News on 25 Apr 1862.
Surmise: Ashley Avenue may take its name from Ashley House and in turn Ashley Drive and Mews take their names from their proximity to Ashley Avenue.
"Resolved – That the name Ashley Drive be approved for a new street off Ashley Avenue on the property of Mr. Joseph McMaster". (18th March 1931).