David's little brother never gets a name in the novel, in part because he lives for so little time that he doesn't truly have a character. The baby is both a plot point, since its birth is what destroys Mrs. Copperfield's health, and a literary device.
As a literary device, the baby allows David to remember his own infancy with his mother, and to think of her as she was before Mr. Murdstone came along and ruined their lives. When Mrs. Copperfield dies, David thinks of his mother as "the mother of [his] infancy; the little creature in her arms was [David himself]" (9.105).
When David attends his mother's funeral and the burial of his baby brother, he is symbolically burying his own early childhood. This scene represents the end of the first stage of David's life and the beginning of his new, lonely, isolated time in Mr. Murdstone's London factory.