Probably named after one of the holders of the title Earl Annesley of Castlewellan, most likely after William Richard Annesley, 4th Earl Annesley (1830–74). He was Conservative MP for Great Grimsby and sat in the House of Lords from 1857 to 1874. Castlewellan Castle was built in his time in 1856-58. The Castlewellan estate was sold by the Annesley family to the government and Castlewellan Forest Park was opened to the public in 1967.
The name first appears in the Belfast Weekly News, September 1864. There are references to people living in Annesley Street (off Carlisle Circus) in The Belfast And Province Of Ulster Directory 1865. In the Belfast Street Directory of 1868 there are 6 houses in the street.