"That on the application of the owner, Mr. R. J. McConnell, four new streets off Ormeau Road, be named Hatfield Street, Farnham Street, Rutland Street and Balfour Avenue".
(17th September 1890).
This street was called after Arthur Balfour, politician, he was in his career amongst other things Chief Secretary of Ireland and Prime Minister. He strongly opposed Home Rule. He is best known for the Balfour Declaration of 1917.
https://www.gov.uk/government/history/past-prime-ministers/arthur-james-balfour