We learned to lip-read, our heads flat on the beds, turned sideways, watching each other's mouths. In this way we exchanged names from bed to bed:
Alma. Janine. Dolores. Moira. June. (1.5-6)
Trapped as they are in the new society of Gilead, the narrator and her peers are forbidden from speaking or even using their real names. Despite that, they find ways to subvert these rules and convey their names, so that they manage at least to preserve this important part of their identities. Fun fact: if you're looking closely, there might be a clue to the narrator's name here. Check out "Character Clues" for more.
This woman has been my partner for two weeks. I don't know what happened to the one before. On a certain day she simply wasn't there anymore, and this one was there in her place. It isn't the sort of thing you ask questions about, because the answers are not usually answers you want to know. Anyway there wouldn't be an answer. (4.18)
This situation, in which one Ofglen is replaced by another, points to the problem of the disconnect between names and people in this society. These women's personalities almost literally don't matter, because they're just replacing each other in spaces where other Handmaids were or are supposed to be.
When I'm naked I lie down on the examining table, on the sheet of chilly crackling disposable paper. I pull the second sheet, the cloth one, up over my body. At neck level there's another sheet, suspended from the ceiling. It intersects me so the doctor will never see my face. He deals with a torso only. (11.6)
Here the narrator's body and mind are divided by a sheet that separates her physical body, the site of potential pregnancy, and her head, where her intellectual self resides. While the narrator is treated as a body that's separate from her self much of the time, here that separation is literalized.
I wait. I compose myself. My self is a thing I must now compose, as one composes a speech. What I must present is a made thing, not something born. (12.25)
The narrator almost seems like she's been split into two parts: within this new Handmaid, "Offred," the narrator works to present a version of her self that is "a thing" she "compose[s]." She can't behave naturally or impulsively; she has to constantly play a role.
My name isn't Offred, I have another name, which nobody uses now because it's forbidden. I tell myself it doesn't matter, your name is like your telephone number, useful only to others; but what I tell myself is wrong, it does matter. (14.37)
Here the narrator tries to distance herself from the new name society has given her. She attempts, unsuccessfully, to convince herself that her name is separate from her identity. Getting to use her "real name" is important: it "does matter." When people are kept from using their real names, they become lesser versions of themselves and start to lose hold of their individuality and uniqueness.
And if I talk to him I'll say something wrong, give something away. I can feel it coming, a betrayal of myself. I don't want him to know too much. (29.16)
It seems here like even the simplest exchanges can reveal portions of identity. The narrator worries that through her conversation she will "give" herself "away," or "betray" what little she has left. While she desires to be known and recognized for herself again, she also has to acknowledge the danger inherent in that kind of action. So much for trying to be yourself.
Falling in love. [...] It was the central thing; it was the way you understood yourself; if it never happened to you, not ever, you would be like a mutant, a creature from outer space. Everyone knew that. (35.13)
While in Gilead it's dangerous to try to remain yourself in any form, in the time before that people believed that "falling in love" was a crucial part of forming one's identity. Now that seems like a luxury that Handmaids living in this new reality can't afford.
I have been obliterated for her. I am only a shadow now, far back behind the glib shiny surface of this photograph. A shadow of a shadow, as dead mothers become. You can see it in her eyes: I am not there. (35.35)
The narrator was robbed of her identity as a mother when her child was taken away, but that didn't mean she didn't feel like a mother anymore. This is a second theft of her maternal identity, when she realizes her daughter no longer remembers her. This makes her feel like she doesn't even exist.
I tell him my real name, and feel that therefore I am known. I act like a dunce. I should know better. I make of him an idol, a cardboard cutout. (41.19)
Here the narrator explicitly connects the power of her "real name" with being known and understood. This calls on the fairy tale trope of a name-giving power, like how Rumpelstiltskin's name is the answer to his riddle. And in fact, admission of the narrator's name to Nick does give him power over her.
"I am Ofglen," the woman says. Word perfect. And of course she is, the new one, and Ofglen, wherever she is, is no longer Ofglen. I never did know her real name. That is how you can get lost, in a sea of names. It wouldn't be easy to find her, now. (44.15)
You know how Descartes said, "I think, therefore I am"? Well, all this new woman has to do is say, "I am Ofglen" and she is—simple as that. This transfer of identities echoes the one at the beginning of the book, when the Ofglen the narrator knew replaced another, initial Ofglen. Similarly, the narrator may be called Offred now, but there was another Offred before her and may be another one to follow.