"On the application of the owners, your committee have named ... four new streets off Ormeau Road Hatfield Street, Farnham Street, Rutland Street, and Balfour Avenue respectively...". (Belfast News-Letter - Thursday 02 October 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