Lois Lowry isn't playing coy with this title—no Shakespeare or Bible allusions here. (Thanks, Lo!)
The term "The Giver" refers to the old man, the former Receiver, who transfers all his memories to Jonas. The very names "Giver" and "Receiver" remind us of one of the book's central themes: memory is meant to be shared.
The professions held by Jonas and the old man may be about holding the memories, but they are more significantly about passing them from one person to another. That's why their titles aren't "Memory Keepers." The transition from one to the other—from a Receiver to a Giver—is an important one in the novel. The old man, of course, becomes The Giver as soon as Jonas becomes the Receiver.
But, more interestingly, Jonas becomes The Giver when he gives his memories to Gabriel. He has found love by the end of the novel, and he expresses that love through the selfless act of handing over his best and brightest memories. Which is totally aww-inducing.