The Avenger is Pip’s servant while he is in London.
Pip decides that as a gentleman he should have a servant. Unfortunately, having a servant is not a pleasant experience for him—thus the name.
I had got on so fast of late, that I had even started a boy in boots—top boots—in bondage and slavery to whom I might be said to pass my days. (ch 27, p. 149)
This is ironic, of course, because the servant is supposed to be the metaphorical slave to the master, not the other way around. This is why Pip calls him the Avenger or phantom. He is revenge for Pip’s living beyond his means and hiring a servant.
Pip finds it difficult to keep a servant because he cannot find anything for the servant to do.
I had to find him a little to do and a great deal to eat; and with both of these horrible requirements he haunted my existence. (ch 27, p. 149)
The Avenger is just another way that Pip finds being a gentleman cumbersome. He feels that he should have a servant to fulfill his role as a gentleman, but he actually does not have anything for the servant to do. He also finds the servant more expensive and time consuming than he thought, because he has to dress and feed him.