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.
This entry for Ashley Drive requires further research to establish when it was officilally named.