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 Bunbury surname in historical dictionaries
Patronymica Britannica (1860)
BUNBURY. A cadet of the Norman house of St. Pierre accompanied Hugh Lupus, earl of Chester, at the Conquest, and obtaming the manor of_B. in Cheshire, assumed his surname from it.