Lower, Mark A (1860) Patronymica Britannica: a dictionary of the family names of the United Kingdom. London: J.R. Smith. Public Domain.
We don't have any stats on how common this name is. This is probably because it's very rare in the UK.
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The Ton surname in historical dictionaries
Patronymica Britannica (1860)
BARRING TON. Some of the families of this name claim a Norman descent, and derive their name from Bareutou (which see). The Irish baronet deduces himself from a Saxon progenitor, keeper of the Forest of Hatfield in the daj'S of the Conqueror. Le Neve derives the name from an imaginary Saxon called Barentine, but according to Sir Jonas Barriugton's Memou'S, the family's Norman origin is unquestionable. The surname was variously VTitten Barentin, Barentyn, Barenton, Barentine, and at length took the English form of Barrington, There are .parishes bearing this name in four English counties.