The name Josette is a girl's name of French origin meaning "Jehovah increases". Josette and discussed in our forums with posts like "Girls Makeover - Two Middles".
From the experts:
One of the group of French 'ette'-ending names that emigrated to this country in the last century, though never as popular as some others, such as Claudette and Paulette--it did have a little six-year flurry here between 1967 and 1973. Josette may be feminine and flirty, but, given the choice, most parents today would prefer Josephine.