Wordorigins.org

This adjective, meaning joyful or light-hearted, is of uncertain origin. The English word comes from the French gai, but where this French word comes from is uncertain. There are cognates in other Romance languages, notably Provencal, Old Spanish, Portugeuse, and Italian, but no likely Latin candidate for a root exists. The word may ultimately be Germanic in origin, with the Old High German wâhi, meaning pretty, and gâhi, swift, being suggested as possible progenitors, but the transition from the medial h in those roots to the Romantic forms is problematic.

The word is first recorded in English sometime before 1310 in a poem found in T. Wright’s Specimens of Lyric Poetry:

Heo is...Graciouse, stout, ant Gay, Gentil, jolyf so the jay.
(He is…Gracious, stout, and Gay, Gentle, jolly as the jay.)