'Finchley' probably means Finch's clearing, or the clearing of the finches.
The hamlet here, first mentioned in 1365, was East End, and the station opened in 1867 as East End, Finchley. Residents asked for a change, to avoid confusion with the poor East End of London, and it became East Finchley in 1886 or 1887. Eric Aumonier's archer on the station, unveiled in 1940, was a metaphor for trains shooting into central London.