Don't know the answer, but seems that it was known as Little Hamilton Street until at least 1925, according to NLS maps: https://maps.nls.uk/geo/explore/#zoom=17.730374944849284&lat=55.86208&lon=-4.24770&layers=109707890&b=1
I'd thought it might be related to nearby Margaret and John Streets, but they were named like that in the mid 1800s.