From seeing how people reach this site, it became clear that there are a lot of people trying to find family in North and South Stoneham. Therefore here is information on where you can find more details. I've also tried the list the churches and other places that may be able to help further with your enquiries.
Very detailed histories of the history of each parish, from the formation of the manors onwards, can be found at British History Online.
South Stoneham and North Stoneham.
Very good, free, early 20th Century maps can be found at A Vision of Britain Through Time.
Neither parish is founded on a village, but from the Manors of North and South Stoneham, which by the mid 19th century were both under the Willis-Fleming family and the histories of the manors can be found on the website of the Willis-Fleming Historical Trust.
The boundaries of both parishes varied over time and their ecclesiastical boundaries could differ from their civil ones, meaning that some of the places below were moved in and out. However, North Stoneham mainly covered the area formed by Chilworth, North Stoneham Park, Bassett Green and Bassett.
South Stoneham covered Swaythling, Portswood, Highfield, Bitterne and West End. It disappeared altogether in boundary changes 1933.
To confuse things even further, South Stoneham registration district (founded 1837) was much larger and included North Stoneham. In 1899 the OS map shows that it included the areas of Chilworth, Bassett, Eastleigh, Swaythling, Mansbridge, Itchen, Highfield, Portswood, Shirley, Sholing, St. Mary Extra, Bitterne, West End, Netley, Botley, Bursledon and Hamble.
In 1927 the civil district of South Stoneham disappeared in boundary changes, becoming parts of Southampton.
Click below for lists of the churches in the parishes:
Churches for the Ecclesiastical Parish of South Stoneham (or appeared within boundaries of the parish at any point).
Churches in the Registration District for South Stoneham