Irish Meaning: The name Mayra is an Irish baby name. In Irish the meaning of the name Mayra is: a Gaelic form of Mary: bitter. American Meaning: The name Mayra is an American baby name.In American the meaning of the name Mayra is: a Gaelic form of Mary: bitter.SoulUrge Number: 2. Expression Number: 22. People with this name have a deep inner desire for love and companionship, and want to work with others to achieve peace and harmony.merican Meaning: The name Mayra is an American baby name. In American the meaning of the name Mayra is: a Gaelic form of Mary: bitter. SoulUrge Number: 2. Expression Number: 22. People with this name have a deep inner desire for love and companionship, and want to work with others to achieve peace and harmony.