'Finchley' probably means Finch's clearing, or the finches' clearing.
It is a late Anglo-Saxon name ending in ley, a clearing, but it was recorded only from the early 13th century and is not in the Domesday Book. West Finchley is a much newer name: the suburb was created in the 1930s between Dollis Brook and the railway, with its own station from 1933.