How does the calculated omission of Miss Brill's first name contribute to her characterization?

The omission of Miss Brill's first name emphasizes how isolated she is from those around her, from society as a whole. She inhabits a world of her own, a world of fantasy. And in this world, her identity is largely self-constructed. The people and the events she witnesses at the park are part of a play in which she has a leading role. As an actress in this human drama, Miss Brill must adopt a completely new persona, one that separates her from her public identity. Her first name is an essential component of that public identity and is therefore irrelevant to the ensuing action. Miss Brill desperately wants to re-establish some connection with the outside world but cannot do so. She can only engage with it by way of fantasy and role play. The omission of Miss Brill's first name is a reflection of that. Just as we will never know her full name, so she will never fully be known or accepted by the people around her.

Miss Brill imagines herself to be quite a proper lady.  She decided to wear her fur, a "Dear little thing!" for her Sunday out to the park to hear the band play.  She likes to sit and watch the people around her, eavesdropping on their conversations, all the while pretending disinterest.  She watches the crowds, couples and groups stopping to talk, some buying handfuls of flowers, and the children, dressed in their Sunday best.  To her, it all seems to feel so bright and cheerful and interesting.  "Oh, how fascinating it was!  How she enjoyed it!  How she loved sitting here, watching it all!"  After she's seen her fill, she typically stops at the bakery on the way home to buy a piece of cake.  All along, she's called Miss Brill and never by a first name: this choice adds to the sense of formality with which she conducts herself.  She never involves herself in conversation with anyone else at the park; she remains apart, aloof.  The deliberate omission of her first name adds to the elegance, with her fur and her almond cake, that Miss Brill seems to believe she possesses. (This is partly why she is so devastated to be the butt of the young couple's jokes.)