"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). The surmise is that the street was named to acknowledge Lord Salisbury who was vehemently opposed to Home Rule for Ireland. There was considerable opposition to Home Rule in Belfast at that time. Lord Salisbury was born in Hatfield House.
https://www.historytoday.com/archive/months-past/retirement-lord-salisbury