There's no definative answer to that question since there are no official records of his name. He came to the world's attention after the storming of the Bastille, when lists and records of the many people who had been imprisoned there were made public. According to the records, a prisoner who was locked up there from 1698-1703 was listed simply as "the man in the mask". He died in prison in on Nov. 19, 1703. (It should be noted that, in the official records, the mask was never stated to be made of iron. That was added later by the writer Alexander Dumas. We don't know what the mask was actually made of.)
According to the journals and reports of the head guard/second in command of the Bastille, M. DeRoarges, the orders were that no one was to speak to him or believe anything the man says. They had orders to threaten him with death if he spoke about anything except his basic needs. The man's name was never revealed to anyone. The guards had orders to shoot him if he ever got the mask off. The man in the mask was always locked up alone, with no company, but he apparantly read a lot.
The man in the mask was in the direct, personal charge of the Bastille's governor Benigne D'Auvergne de Saint Mars. When St. Mars arrived at the Bastille as the new governor, he already had the man in the mask with him. According to the journals of M. DeRoarges, the man in the mask had been in St. Mars keeping for several years, at two previous prisons which St. Mars had been in charge of, before he came to the Bastille. DeRoarges was the only one to ever enter the cell and he took personal care of the man in the mask.
There has been a lot of speculation about who he really was. However, of all the theories, the most likely and generally accepted idea is that his name was Eustace Dauger. Dauger was the valet of influential Lord Roux de Marsilly, who was a political enemy of King Louis and who attempted to begin a Protestant rebellion against Louis. Roux was publically tortured and killed. There's no record of what happened to his valet Dauger, but Dauger may have known the names and faces of his master's co-conspirators. It's possible the king didn't want the public--who were already becoming disgruntled--to know how high the conspiracy went (Louis probably killed all the high-placed conspirators) so he had Dauger locked away and forbade anyone to talk to him.