I don’t know what the origin of this street name is. There are several male Christian names in the area (Charles Street, George Street, James Street). The others have been the forename of one the Dukes of York, which would fit with both York Road and the Duke of York pub on the same estate. However, Sidney doesn’t fit in with this, so I suspect the correlation between Charles, George and James is a coincidence – they are after all fairly common names.
Sidney Street in London was infamous for the siege of Sidney Street in 19111
[...] in the same development are seemingly random forenames (or possibly surnames) – for example Sidney Street, George Street and James [...]
[...] Lane is to the north of Salisbury’s city centre – close to Sidney Street, Coldharbour Lane and Middleton [...]
I think you will find the streets in this area are named after the children of the man who bought the ground after the jail compound was knocked down. Possibly a Mr Scammel, who featured very big in Salisbury’s developement and history. Many of these houses were inhabited by railway workers at the time.