Think Baby Names - Gaius: "... the meaning of Gaius is "happy". Probably the root of the word "gay", which used to mean "jolly"." Credit: Leslie Dunkling and William Gosling, authors of "The Facts on File Dictionary of First Names", Alfred J. Kolatch, author of "Dictionary of First Names", and E. G. Withycombe, author of "The Oxford Dictionary of Early English Christian Names".