(origin: Gaelic.) From Aimhrea (the "mh" having the sound of "v"), denoting contention or disagreement. It may be from Avery, a granary, or from Aviarius, Latin, a bird-keeper.
The English surname Avery is patronymic in origin, belonging to that group of surnames derived from the forename or Christian name of a father. In this case, the surname comes from the Old English "aelf ric," meaning "elf ruler," and is a varient of the Germanic forename Aubrey of the same meaning. The original bearer of this surname would have been the "son of Avery" and would have passed this surname on to his own children, thus making the surname hereditary.
Early records of this surname date back to the thirteenth century, when one Hugo Filius Averey is found in the Hundred Rolls of Nottinghamshire in 1273 A.D., and one Ralph Averey is found in the Rolls for Oxfordshire, together with one Nicholas Auverey in the Worcester Rolls and one William Auure in the Warwick Rolls. In 1279 A.D., one Cust Alvere and one Edmund Avered are mentioned in the Rolls for Cambridge and one Walter Avery in the Rolls for Oxford. This surname first appeared in 1630 when Christopher Avery (a weaver, born in England circa 1590) settled in Massachusetts, bringing his only son, James. His descendants were prominent members of the army and Guardians of the Pequot Indians. In April 1635, one Thomas Avery, aged 18, sailed to the New World. In June 1635, Joseph Averie, aged 20, sailed for Virginia.
Surnames of Anglo-Saxon, Norman French, or Middle English origin.
English - Patronymics
Allen - Alan
Austin - Augustine
Avery - Auvere/Alfred
Avery Formerly a Norman French surname form of Alfred
Avery fairly rare: Ulster & Dublin. English & Scottish. Ir. Mac Aimhréidh, aimhréidh, dishevelled: see Mac Cavera. SI.
from Irish Ancestors section of ireland.com / Sloinnte na hÉireann-Irish Surnames
I'm no expert on the matter, though. Just found it peculiar that for instance immigrant records counts for common Irish surnames vastly outnumbers their UK counterparts (Murphy, Boyle, Finley, Rourke, Ryan), while Irish Averies on the other hand are relatively scarce.
Conclusion: there are three basic scenarios.
1) The name is Old English-Norman in origin. It has spread to Ireland, hencethere are Irish people named Avery.
2) There are (at least) two different origins of this specific surname. One of norman-english and another from gaelic.
3) Sean Avery is in fact, not Irish. (this is unlikely)