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 Amory surname in historical dictionaries
Patronymica Britannica (1860)
AMORY. A:MERY. From the personal name Emeric or Alraericus, equivalent to the Italian Amerigo, latinized Americus, whence the name of the great western continent. It seems to have undergone the following changes: Emeric, Emery, Amery, Amory, Ammorj and in Domesd. Haimericus. It is asserted, however (B.L.G.), that " the fiimily of D'Amery came to Engl, with the Conqueror from Toui-s."