Rosemary is the name of an herb, a natural source, and it is also symbolic of memory. For instance in Act IV of Hamlet, Ophelia scatters flowers about prior to her suicide,
There's rosemary, that's for remembrance. Pray you, love, to remember. (4.5.172-173)
In Chapter 18 Jonas learns about Rosemary, he is told that her name is not supposed to be uttered by anyone. The Giver explains to Jonas that Rosemary was eager to learn, but when he gave her loneliness and loss, she appeared stunned. "But everything changed once she knew about pain," the Giver explains. The times of laughter were gone, then, he adds.
Because she could not stand the memories of pain, Rosemary asked to be released. And, because being released was still permissible, Rosemary was granted her wish. But, when she died, the memories came back to the community since memories are forever, the Giver adds. Some of the memories which returned were important ones and others were burdened with them. As the Giver explains, Jonas perceives the tragic effect that Rosemary's death has had upon him as the Giver speaks with a bitter voice.