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Family name Gayner (Gayner ) , throughout the world, is a rare last name. The family name Gayner is habitual in Bermuda, where it is quite a common surname, Australia, where it is a rare surname, and the United Kingdom. Not in comparison with a country's population, it is commonest in the United States, the United Kingdom, and Australia. Much less frequently, Gayner is the forename as well as the last name.