Probably County Galway based on the surnamedb.com/Surname/Hynes information:
Recorded as Hine, Hyne, Hines and Hynes, this is an English or Irish surname. It has two distinct possible origins, each with its own history and derivation. Firstly, it may be of Anglo-Saxon origin, and an occupational name for a servant in a great household.The derivation is from the Middle English word "hine", meaning a lad or servant and originally a collective term for a body of servants, from the Olde English pre 7th century word "hiwan", meaning a household. Servants in important households were highly regarded, and frequently held senior positions enjoying a number of privileges. Such posts would often become hereditary. Early examples from this source include William le Hyne of Oxfordshire, in the year 1240, and Robert le Hine of Suffolk, in 1273.
The second possible origin is Irish and a form of the pre 10th century Old Gaelic surname O' hEidhin. This was a leading clan of County Galway, and descended from the first chief known as "Guaire the Hospitable", king of Connaught. The clany held the lordship of Aidhne, with Mulroy O'Heyne, who was father-in-law of Brian Boru, being the first to be styled lord of Aidhne in about the year 1010. Early examples of chrch recordings in England include William Hines who was christened at Lavenham, Suffolk,on April 4th 1611, whilst on May 5th 1846, Charles Hines, aged 22, was an Irish famine emigrant. He embarked from Belfast on the ship "Jane" bound for New York, and it is estimated that at least sixteen others of the name followed him in the same year. The first recorded spelling of the family name is shown to be that of Mulroy O'Heyne. He commanded the forces of Connacht, during the battle of Clontarf, in 1014. Throughout the centuries, surnames in every country have continued to "develop" often leading to astonishing variants of the original spelling
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