"The Fourth Amendment (Amendment IV) to the United States Constitution is the part of the Bill of Rights that prohibits unreasonable searches and seizures and requires any warrant to be judicially sanctioned and supported by probable cause."
Fourth Estate seems very plausible. The idea of the 'Fourth Estate' is that the media (the fourth estate) keeps an eye on the government. Whether this is the intention or the 4th Amendment is the intention, the other of the two is a nice coincidence.
When written as one word, "Citizenfour" as in the movie's title (not sure if he actually used it like that), my first thought was possibly a connection to "carrefour," which is French for a road intersection. The 4th amendment theory seems more plausible, but a crossroads provides some interesting imagery in ralation to Snowden's intentions vs the current course of mass surveillance.