Although here in this lovely 1786 map the unwooded parts of both areas are I think included as Westwood Common (presumably the origin of Westwood Park):

http://www.oldlondonmaps.com/cary/cary36.html

Then Forest Hill but no Honor Oak on this 1819 map:

http://www.londonancestor.com/maps/map-leigh.htm

and this 1885 map of the Borough of Camberwell seems to attach the name Forest Hill to the whole area, but Honor Oak appears just as the name of the railway station:
http://www.londonancestor.com/maps/bc-camber-s.htm

Maybe Forest Hill looks like the most enduring name for the whole area, but quite a lot more wobbly and less established than Sydenham or Brockley or Dulwich, by the look of it?