In his book Away Up Thonder Bobby Foster, Shankill local historian states that Ashmore Street was called after the Ashmore family.
Records indicate the “Ashmore family” took on ten building lots in the new Shankill development (an exceptionally high number). In essence, the Ashmores acted as property developers or significant investors in the new streets being laid out on Belfast’s western side. This is confirmed by the Tithe Applotment Books of 1826 for Shankill parish, which record John and Richard Ashmore as substantial occupiers of land in the “New Enclosure” townland . Both “Jno. Ashmore” and “Richd. Ashmore” are listed under New Enclosure in 1826, indicating they held property there. The New Enclosure was a section of the Town Parks west of the old town, precisely the area that would later include new working-class streets off the Shankill Road. The Ashmores’ involvement implies they likely leased plots from the Donegall estate and perhaps built houses or tenements as Belfast’s population boomed. Indeed, between 1822 and 1831 the Donegall estate granted many perpetual building leases to local entrepreneurs , and the Ashmores were clearly among those investing in Belfast’s growth.
"The 1822 Settlement of the Donegall Estates" by W. A. Maguire in Irish Economic and Social History Vol. 3 (1976), pp. 17-32.