When he was in the possession of Lacie, Oz was never given a name for whatever reason. However, once in the possession of Alice, he obtained the name "Oz," the name we know him by. (Alice named him this after Oswald.)
So, to answer your question, Oz doesn't really have a real name. His name is just Oz. He's also called the Bloody Black Rabbit/B-Rabbit, but that isn't his real name.