I know a French lady named Josette. She told me that her understanding is that her name is a diminutive of "Josephine", once a very popular name in France (Napoleon's empress).
Josette may also find its origins in the name Cosette, from Victor Hugo's "Les Miserables", as a sort of play on words. If one accepts the idea that Cosette is derived from Chosette, it's rather easy to see the leap from Chosette (shoh-zet) to Josette (zhoh-zet). The "play on words" comes in if you imagine combining the "jo" of Josephine with the "-osette" of Cosette, effectively marrying one image of the French feminine ideal (quiet regal strength) with another (vulnerable innocence).