Only the family and intimate friends know. If I were one of them, I would call her really name, "steph". GaGa would only be her name when refering to pop music industry. They give her this name and play such tricks to promote business. However, I may be wrong.
On the Jonathan Ross talk show in April 2009, Lady Gaga claimed that her family and friends call her "Gaga." However, I think that was more of an act; she was a new artist at the time, and was most likely trying to commit to the character of Gaga she had created. I would think her family/close friends call her by her given name (or some nickname), Stefani.