Locals know the wood as Bourne Wood, or The (Lower) Bourne. The name "Sable Wood" was coined by Woodlands.co.uk when the wood was put up for sale in September 2011. There doesn't appear to be any historical background to this choice of name, but since sable is French for sand and the site is characterised by sandy soil, it does seem appropriate.
To give an absolutely complete picture, the wood was in fact bought as two 3 acre plots, with Woodlands.co.uk acting as agent in both cases. The first parcel, comprising the land east of the central footpath was Sable Wood. The second parcel, to the west of the path, was given the name Spire Wood by Woodlands.co.uk. There seemed little reason to maintain two names and the name Sable now applies to the whole area.