It seems likely that this street was named after one of the holders of the title "Earl of Onslow", a title in the Peerage of the united Kingdom created in 1801 and still in existence. This argument is strengthened by the presence of Haddington Gardens and Shelbourne Road in the same neighbourhood. Haddington (East Lothian, Scotland) and Shelburne (sic, Co. Wexford, Ireland) are unrelated geographically but are also names of earldoms. The title "Earl of Onslow" relates to Onslow near Shrewsbury in Shropshire, England. Onslow Hall (demolished) was once the residence of the Onslow family, but the seat of the current Earl of Onslow is Clandon Park in Surrey.
"Resolved — That on the application of the owners, Messrs. The Martin Estate Co. Limited, a new street on their property, situate off Cregagh Road, be named Onslow Gardens". (13th June 1911).