This was the name of the first English baby born in the New World: Virginia Dare in 1587 on Roanoke Island. Perhaps because of this, the name has generally been more popular in America than elsewhere in the English-speaking world, though in both Britain and America it was not often used until the 19th century. The baby was named after the Colony of Virginia, which was itself named for Elizabeth I, the Virgin Queen. A more recent bearer was the English novelist Virginia Woolf (1882-1941).
The town's name is said to have been derived from Mazzara, which either originates from Arabic mazari or mazar meaning "shrine", or is a corruption of Mactorium, the name of a town that had existed in the area in ancient times. That town had been founded by ancient Greek colonists, who had called it Μακτώριον (Maktorion). It is uncertain what the town's name meant in Greek, but it is possibly related to the Greek noun μακτήριον (makterion) meaning "food". Also compare Μαιμακτηριών (Maimakterion), which is the name of one of the lunar months of the Hellenic calendar used in ancient Attica. Alternatively, an etymological relation with the Greek adjective μακτός (maktos) meaning "kneaded" is also possible. This word is ultimately derived from the Greek verb μάσσω (masso) meaning "to knead, to press into a mould".
With that said, the given name Mazarine is quite rare in France today. It was virtually unknown in the country, until the existence of Mitterrand's illegitimate daughter was revealed in 1994 or 1995. The name gained quite a bit of exposure after that, which made it possible for prospective parents to take a liking to the name and bestow it upon their daughters. This clearly shows in the available statistics for the name Mazarine: it suddenly appeared on the radar in the mid-1990s and has remained on it ever since, whereas in previous decades, the name was not used on a noteworthy scale at all (as was to be expected, since this name was more or less "invented" as a given name).