Her nickname comes from the 1959 book "Zazie dans le métro" by author Raymond Queneau (there's also a movie of the same title by Louis Malle).
Rue de la Paix - Zazie
Zazie - Des Rails
By the way, the "Rue de la Paix" title is a reference to the most expensive property to own when playing the French monopoly. She's actually saying "Mettre un hôtel Rue de la Paix" in the chorus, which means, "Put/place an hotel on Rue de la Paix". don't know, maybe it's good to know that he 2 most desirable streets to own in French monopoly are the Champs-Elysées and Rue de la Paix, the latter being slightly more expensive and right next to the starting space. They correspond to Park Place and Boardwalk in the American version of the game.