'Balham' probably means a homestead in a rounded enclosure.
The place is in the Domesday Book of 1086 as Belgeham, held by Geoffrey Orlateile. 'Bal' refers to a rounded enclosure, and 'ham' to a homestead, village or river enclosure. Maps of the 1600s show it as Ballam, Balham Hill or Balham Manor. The first railway station, opened in 1856, was called Balham Hill.