Novels that occur in dystopian futures typically use unique — and sometimes silly — names to signify that this world is unlike the one people live in today. For example, Stephen King took issue with Katniss' name in his review of The Hunger Games in 2008, writing, "lame name, cool kid" and "once I got over the main character's name (Gale calls her Catnip — ugh), I got to like her a lot." If you haven't read the book it's based off, but know that Elisabeth Moss' character's name is Offred in The Handmaid's Tale series on Hulu, you might be having the same feeling of aversion that King had toward the names in Suzanne Collins' YA book. Yet, the names of Margaret Atwood's characters are not bizarre just for the sake of it — the meaning of the name Offred in The Handmaid's Tale actually indicates the character's position in the Republic of Gilead.
The Handmaid's Tale takes place in the future where a theocratic regime rules the former U.S. and has stripped women's rights based on lessons from the Bible. Infertility has plagued this new world and women who are still fertile are forced to become "handmaids" and must have ritualized sex with men in leadership positions ("commanders") to produce children from them and their wives. As the handmaids had independent lives before Gilead was formed, these women did have other, real names in a past life when they weren't enslaved to the male commanders. (Offred's former name is never explicitly stated in the book.) All of this background information is necessary in order to fully understand the horror of what Offred's name means.
In the book, the handmaids all have names that relate to the men they serve, examples being Offred, Ofglen, Ofwarren, Ofcharles, and more. Moss' character Offred is the handmaid to Joseph Fiennes' Commander Fred Waterford and as she is tasked with producing a child for him and his wife, her name literally is "of Fred." This naming convention is the same for all handmaids as Alexis Bledel's character Ofglen serves Glen.
Beyond the name Offred indicating that she is a slave "of Fred" and has no identity beyond bearing him a child, it's no coincidence that her name is a play on the word "offered." As Atwood wrote in The New York Times for a March 2017 article about her 1984 novel:
Unlike The Hunger Games' Katniss, Offred in The Handmaid's Tale has this unconventional name not just because the book takes place in the future, but to show the patriarchal world the handmaids belong to. So if you weren't horrified enough by the future that The Handmaid's Tale represents, the naming convention that Gilead uses shows how far this dictatorship will go to control women — stripping them completely of their own identities so that even their names prove they only exist to serve men.