In choosing a motto in French rather than English, it should be noted that the English language had only recently replaced French as the language of the English ruling classes - and Henry spoke French and had been crowned King of France as well as of England. The motto of the Order of the Garter Honi soit qui mal y pense is also in French.
The phrase was allegedly first used as a password by King Richard I in 1198.
Dieu et mon droit was the inspiration for The Beatles joke motto Duit On Mon Dei, later adopted as an album title by Harry Nilsson.