One of the themes of this story revolves around urban society in that late 19th century. It was important then, as it often is now, to conform to a certain role in order to be recognized. Once identified in that role, society knew what to expect of you. Often, that role centered around work. This is certainly true in the case of the narrator. He is unnamed because it is his position, as "the lawyer", that identifies him. He speaks of nothing outside his work - his concern with Bartleby centers around the fact that Bartleby is his employee. Nothing is ever mentioned of his home life, family, etc..
Turkey, Gingernut, and Nippers, sharing similar positions, have to be identified by personality traits. Again, names have lost importance, because names suggest individuality. These nicknames, which center around how these characters behave at work, are used instead. Turkey is so known because, as is suggested, he indulges himself too much at lunch, causing his afternooon work to suffer. Nippers is so named because he is irritable and seems unsatisfied with his position in society. Gingernut earned his name because off the treat he brings back when he is sent on errands. Again, for all these characters, no personal life is mentioned. Melville's theme is that workers in America have no personal life, no individuality - which is what brings Bartleby's downfall in the end.