Why doesn't Katherine Mansfield give "Miss Brill" a first name?

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.

Why is the name "Brill" chosen for the character in Katherine Mansfield's "Miss Brill"?

I would add to the existing educator comments the fact that, while "Brill" is a real, although relatively uncommon, name in the United Kingdom, the word "brill" at the time of Mansfield's writing would have been far more commonly heard in another context—that of the brill fish. Mansfield's short story predates the use of "brill" as a common shortening of "brilliant," although the use of the word "brilliantly" in the opening line of the story does invite an association between this and Miss Brill's name. It is likely that the audience would have made a connection between Miss Brill and the North Atlantic flatfish brill, similar to turbot, which appears with frequency on menus in the United Kingdom.

This, in connection with the "motionless" air, the "faint chill" and the fact that Miss Brill is "glad that she had decided on her fur" leads to an association between Miss Brill and a "cold fish"—that is, someone both physically and emotionally cold or reserved. The idiom "cold fish" actually comes from Shakespeare, but finds regular application in the modern day and means an unfriendly or apparently unemotional person, often a woman. Whether or not Miss Brill genuinely is this kind of woman, the choice of this name for her may be an indication from Mansfield of how she is generally perceived.

When Miss Brill is introduced, in the first paragraph, the word "chill" is repeated twice in the very next sentence. Brill is a somewhat unusual name, and so the sound similarity—the fact that it rhymes with chill—seems not coincidental. This association appears to be appropriate because of Miss Brill's own sort of coldness, the fact that she doesn't really engage with anyone else around her. In fact, the narrator says, "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 is not a warm person who invites others to make conversation with her; she doesn't reach out to anyone else the entire time she's at the park. She is barely even noticed, and the one time she is, it's because she's being made fun of. In fact, almost the only personal human contact she has is when she reads the newspaper to an old man four afternoons a week, but even he always sleeps through her visits. Miss Brill's life is a chilly one, indeed. Despite her fantasy to the contrary, there are few, if any, who would miss her, and the similarity of her name to the word "chill" helps to draw attention to this connection.

You are right in indicating that the name "Brill" seems to suggest the adjective "brilliant." My own thoughts on this matter is that the name that Mansfield gives to her protagonist reinforces the essential divide that exists between Miss Brill's own perception of her life and the reality of it. There is therefore immense irony in giving Miss Brill a name that suggests the adjective brilliant, as it is obvious to the reader that her life is anything but.

Note how Miss Brill views her own existence as something extraordinary and dramatic. She spends lots of time indulging in her fantasies that imbue her with an importance that it is all to evidenct she lacks in real life. Consider the following quote:

Even she had a part and came every Sunday. No doubt somebody would have noticed if she hand't been there; she was part of the performance, after all.

In her lack of perception, her life is "brilliant," but as we continue to read the text we see the reality that Miss Brill is either not aware of or chooses not to see. The central irony of the text is that Miss Brill observes others and thinks that they had "just come from dark little" cupboards when she herself is shown to retreat to "her room like a cupboard" at the end of the story when she has her illusion punctured by the rude comments of the couple. Thus the name of "Brill" heightens the disparity between reality and perception and reinforces the theme of lack of self-perception.

Why doesn't Mansfield give Miss Brill a first name?

Miss Brill is a very lonely woman, so lonely that “when she breathed, something light and sad–no, not sad, exactly–something gentle seemed to move in her bosom” just by putting on her fur collar. In fact, her fur collar is her best friend. Rather than participating in conversations, she instead eavesdrops on those of others, “sitting in other people's lives just for a minute while they talked round her.”Such women in our society are considered odd, to the extent we think of them at all, and certainly do not get to know them.  By telling the story from third person limited point of view of Miss Brill, Mansfieldallows us into this lonely woman’s world, just a little.  However, just like everyone else, her narrator stays removed from her character, although she presents her with compassion and respect. Those qualities are best conveyed through the formal “Miss Brill” rather than speaking about her intimately through a first name, for surely no one is on a first name basis with this woman.