What is the new Mrs De Winter's first name?
Soon after Maxim and the narrator meet, he comments on her Christian name, saying that it is 'lovely and unusual'. She says that the name was given by her father, who was an artist, and that he was 'a lovely and unusual person'. Her father's death, and her being left alone with no assets or income, is the main reason she is now working as a 'paid companion' to the horrible Mrs van Hopper. When he sends a note of apology following their first meeting she comments that her name is on the envelope, and it is spelled correctly, 'something that hardly ever happened', which also marks Maxim as someone who is educated, sensitive, and in tune with her in some way.
So, although we never learn the narrator's first name, we do know that it's 'lovely and unusual' and that Maxim obviously likes and recognises it. Additionally, we can see that her father's contribution to her meeting Maxim de Winter takes a number of forms: he gives her a name which attracts him; his death has led to her presence in his upper class world; and it could also be argued that having had a very close relationship to her father, now dead, she is vulnerable to the offer of another older man to 'look after' her.
As the previous answer states, du Maurier never names Maxim's second wife--the narrator. Her purpose in doing so is mostly likely to lend credibility to the narration--we don't refer to ourselves in the third person; so a narrator realistically would not either. Additionally, the novel has Gothic and mystery elements, and the unnamed narrator technique certainly adds to du Maurier's style (Poe's narrators are often unnamed).
One last point, the narrator begins the book and spends much of the novel in the shadow of Rebecca, her husband's first wife. It is not until the end, that she finds a sense of identity and belonging; so du Maurier's choice to allow her to remain nameless advances the theme of insecurity and isolation (much of which is caused by Rebecca's lasting impression on seemingly everyone she met).
The narrator of the Daphne du Maurier novel, Rebecca, is an orphan whose parents died about a month apart. After being employed by a rich American living in Monte Carlo, she meets the widower Maxim de Winter. Only after she marries de Winter does the narrator receive a name in the novel. However, as the second Mrs. de Winter (the title character, Rebecca, was his first wife), she is never identified by her first name. This remains a mystery to the reader throughout the novel. Maxim is later accused of murdering his first wife.