The answer to this question can be found at the end of chapter 14. The legend of Maniac has gotten fairly large by this point in the story. Jeffrey has been with the Beale family for quite some time, and people are starting to realize that Jeffrey is the Maniac from earlier in the book. Mrs. Beale eventually asks Jeffrey point blank if he is or isn't "that Maniac."
The narrator tells readers that Jeffrey answered her the same way that he answered everybody else:
"I'm Jeffrey. You know me." Because he was afraid of losing his name, and with it the only thing he had left from his mother and father.
Mrs. Beale instantly understands, and she agrees that Maniac will be Jeffrey in her house; however, she points out that she can't guarantee that from the wider community. Jeffrey understands what Mrs. Beale means too. He knows that nicknames are given and can stick no matter how much the nicknamed person may not want to be called by the nickname.
Jeffrey desires to be known by his birth name because it is the name that his parents gave him. It is the only thing he has left from his parents. His parents were tragically killed, and Jeffrey was shipped off to live with his aunt and uncle. For 8 years, he lived with them and their failing marriage. Eventually, he couldn't take it anymore, and he ran away. Any of the clothes and toys that his parents may have given him are long gone. He only has his name, and that name ties him to his loving parents. Maniac doesn't want to lose that connection, and that is why he insists on being called Jeffrey.