This building could be seen in 1901 map, between Church Mission House and Fairlea. The 150th anniversary commermoration publication of St. Stephen's Church (Chinese only: 猜情尋 - 香港聖公會聖士提反堂一百五十周年堂慶歷史相片集) suggests that it was a Women's Hostel (婦人館).
Could this building have been used for other purposes as well? There were indications that it housed the Victoria Home from 1887 to 1902 (SCMP 1936-10-24, p. 12: "The first Victoria Home was in a building next door to St. John's Hall near the University"). The same report suggested that it housed the Boys' Club in the 1930s ("It is now used as a Boys' Club").
In the Jubilee edition of The Chimes (St. Stephen's College publication), it was also suggested that the building housed its science building and tiffin room from 1924 to 1929 (during which the College had to leave its Bonham Road campus due to damage by 1918 earthquake): "The Women's School, next to St. Stephen's House was rented as a Science Building, and as a tiffin-room".
Demolished in c. 1950 to give way for the old St. Stephen's Church Annexe, completed and opened on 28th May 1950.