Like Salisbury Street off Dublin Road, Salisbury Avenue is named after Robert Gascoyne-Cecil, 3rd Marquess of Salisbury, a Conservative statesman who was vehemently opposed to Home Rule for Ireland (John J. Marshall, Belfast Telegraph 03/03/41). He was Prime Minister of the United Kingdom for three separate periods for a total of over 13 years between 1885 and 1902. His three terms as Prime Minister were interrupted by Liberal governments led by Gladstone. There was considerable opposition to Home Rule in Belfast at that time.
The southern section of this street was originally called Fortwilliam Street. It was then changed to Kenbella Avenue in 1876. Somewhat confusingly, the northern end was named North Circular Road, although there is no overlap between that section of road and the current North Circular Road. The name of the middle section was changed again in 1890 to Salisbury Avenue, and in 1895 this name was adopted for the street's entire length.
There was, in fact, a proliferation of Belfast street-names honouring the Marquess of Salisbury at the end of the 19th century. As well as Salisbury Street in BT7 (and the adjacent Salisbury Court, created later), there was a second Salisbury Street in Ligoniel, which was later renamed Lever Street to eliminate the duplication of the name.