Area names in British cities are mostly remnants of the villages and small towns that have been swallowed up by the expansion of adjacent cities. Some of them still retain village-like characteristics.
Areas of London, such as Chelsea, Soho, Westminster etc. are all good examples.
Other areas in other towns are called after the geographical features which used to be their way they were referred to before the brick and concrete arrived. Goldenhill (Stoke on Trent) or Oakwood(Leeds) are two like that. Less often, areas are called after their aristocratic former Landlords or after their home estates. Thus Carnaby Street and Burlington arcade in London have names that refer to a little fishing town in East Yorkshire, which is where their former owner's aristocratic title originated- Bridlington.
British names are history. There is a place near me called British Camp.
The British bit refers to the Britons who resisted the Roman occupation two thousand years ago.
cheers
Dave