Named after Wandsworth Road, which runs from Vauxhall towards Wandsworth.
Wandsworth appears in the Domesday Book of 1086 as Wandesorde and Wendelesorde, meaning the enclosure of a man named Waendel. The River Wandle bears the same name: it is thought to be a back-formation from Wandsworth, although Wikipedia's article on the town says the town takes its name from the river.