Mrs. de Winter's narrative is a personal reflection, taking place inside her mind. So, when she hides her first name, she's hiding it from herself. And it's not that she doesn't remember it, or that nobody calls her by it. (We mean, really, have you ever gone more than a few days where no one said your name to you?) Our narrator clearly and deliberately edits her name out of the story.
Take a look at what she thinks about when Maxim sends her a note to Mrs. Van Hopper's Monte Carlo hotel room: "But my name was on the envelope, and spelt correctly, an unusual thing" (3.50). Some of this seems to be pure teasing, on the part of the author, Daphne du Maurier. She does it again not long after, in a conversation between the soon to be Mrs. de Winter and Maxim:
"You have a very lovely and unusual name."
"My father was a lovely and unusual person."(4.33-34)
Wait a second: is it because of her family that our narrator hides her name? Consider this reflection from the same scene: "It was not easy to explain my father and usually I never talked about him. He was my secret property. Preserved for me alone […]" (4.36). Hmm. It's possible that her real name reminds her of even more painful memories, of being in a loving home with loving parents, and then watching them die within five weeks of each other leaving her homeless and destitute.
Daddy issues or not, we know that our narrator feels an extreme need to be identified as Mrs. de Winter. By editing out her first name, she even forced Shmoop to refer to her as Mrs. de Winter for most of our discussion of her. And of course, her missing name is also a sharp contrast to Rebecca's name which pervades every aspect of the story, including the title.
One last thing: the name "de Winter" can't pass without comment. Most of the story is told during the summer, and there are constant references to the heat. But the name reminds us that there is always the wintry chill of murder and deceit in the air. Fancy.
Rebecca is loaded with conversations where Mrs. de Winter is being asked if she hunts, plays games, plays cards, or sails. Of course, she doesn't, and, of course, Rebecca did. So it seems like the things you don't do matter just as much as the things you do.
Mrs. de Winter has some interest in sketching, which is also a frequent topic of conversation. However, after becoming dissatisfied with her first sketch of Maxim, we don't see her lift pen to page. In this way, she's kind of foil to her author, Daphne du Maurier, who couldn't stop creating art, and to Rebecca, who was incredibly active. Of course, maybe Mrs. de Winter's art is in her own mind, in her dreams, her fantasies, and her memories.
It's not entirely clear what Maxim does all day, but it's probably wheeling, dealing, schmoozing, and politicking. But, his most stunning action, murdering his wife, is what most defines him. It's really hard to try to analyze him without taking his crime into account.