Question: What is the old Irish meaning of the name of the county where Hugh Hamilton was born?

Context paragraphs:
Lillias Hamilton: Dr Lillias Anna Hamilton M.D., (7 February 1858 – 6 January 1925) was a British pioneer female doctor and author.  She was born at Tomabil Station, New South Wales to Hugh Hamilton (1822-1900) and his wife Margaret Clunes (née Innes).  After attending school in Ayr and then Cheltenham Ladies' College, she trained first as a nurse, in Liverpool, before going on to study medicine in Scotland, qualifying as a Doctor of Medicine in 1890.

Conn (name): Conn is a masculine English- and Irish-language given name, as well as an English-language surname.  The origin of the given name is uncertain.  It may be related to the Old Irish "cond" ("intellect" or "mind"), or perhaps to the Old Irish "cenn" ("head", "chief").  The latter word-origin may have resulted from a popular, but incorrect etymology, applied to the Old Irish terms "Leth Cuinn" and "Dál Cuinn"; these terms originally meant "half of the chief" and "tribe of the chief" but were mistakenly regarded to mean "half of "Conn"" and "tribe of "Conn"".  In some cases the given name is as a short form of non Irish-language names that begin with the first syllable "Con-" (such as the names "Conor" and "Connor").

Kjartan: Kjartan (] ; ] ) is a masculine given name found in the Nordic countries, and is most prominently in Iceland and Norway.  The Old Norse name "Kjartan " was a shortening of "Mýrkjartan ", from Old Irish: "Muirchertach" , the name of an Irish king whose daughter Melkorka (Old Irish "Mael Curcaigh ", "servant of Curcach") was brought to Iceland as a slave.  The Irish name Muirchertach, meaning "mariner", is modernised to "Muircheartach ", anglicised as Murtagh.

Gurteen Beach: Gurteen Beach ("Trá na Feadóige" in Irish meaning "beach of the plover"), located in Roundstone, County Galway, in the Connemara region of the west of Ireland, lies back-to-back with Dog's Bay.  The two beaches were formed by a sand spit and tombolo which now separates the two bays known as Gurteen Bay and Dog's Bay.  The name Gurteen derives from the Irish "Goirtín" meaning" small plot" or "small field".

Tyrone: Tyrone (Irish: "Tír Eoghain" , meaning "Land of Eoghan" ) was a kingdom of Gaelic Ireland, associated geographically with present-day County Tyrone, County Armagh and parts of County Londonderry.  The kingdom represented the core homeland of the Cenél nEógain people of the Northern Uí Néill and although they ruled, there were smaller groups of other Gaels in the area.  One part of the realm to the north-east broke away and expanded, becoming Clandeboye, ruled by a scion branch of the Ó Néill.

Hugh Hamilton (racing driver): Hugh 'Hammy' Caulfield Hamilton (18 June 1905 - 26 August 1934) was a British racing driver, who was killed in the 1934 Swiss Grand Prix.  He was born in Omagh, in Tyrone, Northern Ireland.  It has been said that he was "perhaps the top British driver at that time [before his death]".  He raced with Dick Seaman on occasion, and was good friends with him.

Ballyclerahan: Ballyclerahan (Irish: "Baile Uí Chléireacháin" ) is a village in County Tipperary, Ireland.  It is also a civil parish in the barony of Iffa and Offa East.  It is approximately 8.5 kilometres north of Clonmel.  Its name comes from the Irish meaning O'Clerahan's town.  Locally, the "Bally" prefix is usually dropped and the village is almost always referred to simply as "Clerahan" which is most often spelled "Clerihan".

Fearn (letter): Fearn is the Irish name of the third letter of the Ogham alphabet, ᚃ, meaning "alder-tree".  In Old Irish, the letter name was Fern, which is related to Welsh "gwern(en)".  Its Primitive Irish root was *"wernā" and its phonetic value then was [w].  Its Old Irish and modern phonetic value is [f].

Old Irish: Old Irish (Old Irish: "Goídelc" ; Irish: "Sean-Ghaeilge" ; Scottish Gaelic: "Seann Ghàidhlig" ; Manx: "Shenn Yernish" ; sometimes called Old Gaelic) is the name given to the oldest form of the Goidelic languages for which extensive written texts are extant.  It was used from 600 to 900.  The primary contemporary texts are dated 700–850; by 900 the language had already transitioned into early Middle Irish.  Some Old Irish texts date from the 10th century, although these are presumably copies of texts composed at an earlier time period.  Old Irish is thus the ancestor of Modern Irish, Manx, and Scottish Gaelic.

Kevin: Kevin ( ) is the anglicized form of the Irish masculine given name Caoimhín (] , Middle Irish: "Caoimhghín" ] , Old Irish: "Cóemgein" ] , latinized "Coemgenus"), composed of Irish "" (Old Irish "cóem") "dear; noble" and "-gin" (Old Irish "gein") "birth".

Supporting paragraphs: Hugh Hamilton (racing driver) Tyrone
Answer: Land of Eoghan