I think if "Miss Brill" were written today, nothing would have to be changed. The story works then and now the same, because its central character is an out of fashion old lady who people-watches in a public park. I believe that initial set up could take place anywhere at any time.
The story does take place in Paris, but I keep imagining the location as something like Central Park in New York. That would be my modernized setting.
In the story, Miss Brill is wearing a fur. It sounds innocuous enough until you read closely.
She had taken it out of its box that afternoon, shaken out the moth-powder, given it a good brush, and rubbed the life back into the dim little eyes. "What has been happening to me?" said the sad little eyes. Oh, how sweet it was to see them snap at her again from the red eiderdown!... But the nose, which was of some black composition, wasn't at all firm.
The "fur" is an intact animal. Its head is still attached. Miss Brill is walking around with a stuffed, dead animal around her neck. I'm sure at one time that was in fashion. Still, that's gross.
To modernize it, I wouldn't change what she is wearing, but I would change the location and type of dead stuffed animal. I would have Miss Brill carrying a stuffed, tiny dog. She would be carrying the dead dog with one of those doggy purses. See attached picture. That would allow her to think she is fashionable, when she really isn't.