The book: Atwood never reveals Offred’s actual name, though there has long been a theory that it might be June, one of the names she lists when she is talking about how during their indoctrination at the Red Center, she and the other Handmaids-to-be “exchanged names from bed to bed: Alma. Janine. Dolores. Moira. June.” All of the other names show up as characters in the book, leading to the speculation that June is Offred’s true identity.
The show: At the end of the first episode, Offred reveals her secret, forbidden name via inner monologue, and it is indeed June.