Shaw’s Road described in this way in 1932 Belfast Street Directory: Falls Road to Glen Road with eight numbered addresses and “Kilmona” – Magee, James.
Whilst this may be the earliest entry for Shaw's Road in its own right, there are earlier references to Andersonstown from 1918 which mention Shaw's Road as intersecting. None of the early residents named in these directories are named Shaw, so the origin of the road-name is unclear.
The name could be considerably older as the road was marked (but unnamed) on the 1st edition Ordnance Survey 6" map made in the 1830s. At that time Stewartstown Road branched off Andersonsonstown Road, whilst Shaw's Road was the continuation of the main road. The junction was later reconfigured so that Shaw's Road branched off the main road.
Shaw's Road is remarkable for the establishment of an Irish-language community here in 1969 called Gaeltacht Bhóthar Seoighe or Pobal Feirste, and known colloquially in English as "the Irish Houses".
Note the use of Seoighe to gaelicise the name Shaw in this instance, a form usually reserved for Joy and Joyce. The Irish version of Shaw Street in Dublin is Sráid an tSeáigh (https://www.logainm.ie/download/dcc-sraidainmneacha-bhaile-atha-cliath.pdf).