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.
'Brill' is the name of a common deep-sea flatfish having the ability to match its colour to the surroundings.