Melville chooses the lawyer who hired Bartleby to narrate Bartleby's story. The lawyer writes in the first person. I believe Melville made this decision because he wanted to convey an outsider's view of Bartleby, in order to highlight the mystery of why Bartleby behaves the way he does.
The lawyer is a seemingly level-headed person who views Bartleby with sympathy and wants to help him. Still, he never truly gets inside of Bartleby's head. At best, he can only guess what caused Bartleby to say, "I prefer not to" in regard to not working.
A third person narration would potentially allow omniscience, which would make it possible to explain why Bartleby did what he did. So would a first-person narration from Bartleby's point of view. But Melville wants to leave us with a sense of mystery that will encourage us to think for ourselves, the lawyer is the perfect vehicle for this. We identify with him as he tries to unravel the "why" behind Bartleby's self-destructive passivity. He comes up with the Dead Letter Office as a possible explanation. But, in the end, the lawyer is simply an ordinary Everyman, leaving us as readers to grapple with Bartleby in our own ways.
One of Melville's themes in the story is the essential mystery of human behavior, and the lawyer's limits in only being able to see Bartleby from the outside help to reinforce this theme.