Are you sure "Master" is part of the name? Until the middle of the 20th century, it was common for "Master" to be used as the title for boys before they became adults; it was only after one became considered an adult that this changed to "Mister". The exact age at which the title changed of course varied from place to place and time to time, but certainly most boys under the age of 14 would have been called "Master" in the UK and USA in 1880.
Early is an English surname. It is fairly common in Virginia, and I have known men in the South who had Early as a first name as a transfer from the surname. (It's also possible that Early could occasionally have been a pet form of Earl.)
So I would assume that "Master Early Chamberlain" was a boy 14 years old or younger named Early Chamberlain who was one of the first owners or readers of the book in question.