Miss Brill does not have a first name because she is alienated from society.
By never giving the title character a name, the author helps to establish her anonymity and isolation. She is a lonely woman who does not make any actual connection with another person in the story. She has no first name, and therefore never completely makes a connection with the reader either.
Although Miss Brill watches other people, she does not really interact with them.
… Miss Brill always looked forward to the conversation. She had become really quite expert, she thought, at listening as though she didn't listen, at sitting in other people's lives just for a minute while they talked round her.
Miss Brill listens to their conversations, but does not try to talk to them. She is disappointed when they don’t talk, because she wants to see people interact with others. However, she is unable to make this connection herself.
By presenting the character's thoughts but never really describing her, the author creates a distance between the reader and the title character. We know what she thinks, but not really who she is. We only truly see her through others' perceptions of her, or at least those she notices.