The question is not, "What does Bartleby want?" It is never clear that he wants anything at all. The problem all started here:
In this very attitude did I sit when I called to him, rapidly stating what it was I wanted him to do — namely, to examine a small paper with me. Imagine my surprise, nay, my consternation, when, without moving from his privacy, Bartleby, in a singularly mild, firm voice, replied, “I would prefer not to.”
So it's not what Bartleby wants; it's all about what he doesn't want.
There is no question that something is seriously amiss with Bartleby the scivener. But, in this very Kafkaesque tale of the world of business, is there something wrong with Bartleby or is there something wrong with the whole world Bartleby happens to inhabit?
Consider his life. He writes and writes and writes. And what he writes is all copied as if by a machine. Nothing original comes from Bartleby. Nothing. The story takes place before carbon copies, Xerox machines, or computers, where forms had to be hand written and reread over and over and over. And Bartleby plainly states, “I would prefer not to” again and again and again. He shuts down and just refuses.
What does Bartleby want? He wants not to.
He is the reverse image of people who go to work every day at jobs they don't care about doing repetitious, meaningless things and never question what they are doing with their lives for a second.
I think it is pretty obvious what Bartleby wants, since he is such an obstinate character that he ends up getting what he wants. He wants to do exactly as he is doing. He is a loner, an introvert. No doubt he engages in some kind of mental activity, but we cannot tell whether he is thinking, remembering, or fantasizing. He likes the kind of routine work he does because it does not require much mental effort, but at the same time it provides him with an income which enables him to live his quiet life. He wants to be left alone. He enjoys being alone with his own thoughts. He does not need other people, and the people he has to associate with at the office, including the narrator himself, probably seem as eccentric to him as he does to them. There are people in the world like Bartleby--although there are few who are such extreme cases of introversion and introspection. Melville deliberately contrasts Bartleby with the narrator, who is a good-natured, convivial, tolerant sort of man. Melville also sets his story in the busiest part of New York in order to accentuate Bartleby's isolation.