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 Cot surname in historical dictionaries
Patronymica Britannica (1860)
,'COTT. COT. COTE. A common termination of local surnames, as in Walcott, Caldecott, Norcot, Northcote, Southcote, &c. It appears to be the A- Sax c6te. Professor Leo observes that, " a seiche the dwelling of the wealthy — of landowners, cote on the other hand indicates the abode of the poorer classes. Cote is the house of an indigent dependent countryman, who, without any personal estate, holds a transferable tenement in fief. It was originally a house of mud, or of earth, with loam walls." The prefixed word sometimes indicates the owner's name, and is sometimes descriptive of the situation.