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.
Ashley Avenue may have been named from Ashley House, but the reverse is also possible. A connection with Lord Ashley seems likely. In 1857 Lady Harriet Chichester, the only daughter of the 3rd Marquess of Donegall married Anthony Ashley-Cooper, then styled Lord Ashley. He subsequently became the 8th Earl of Shaftesbury and inherited Belfast Castle.
Ashley Mews was named considerably later in the 1970s, presumably from its proximity to Ashley Avenue.