Here is the story that explains how the cranes relate to the name of the quilt:
The connection between origami (folded-paper) cranes and the hope for peace came about because of a young Japenese girl called Sadako Saski, who lived in Hiroshima. She was only two years old when the atom bomb was dropped on her city. Sadako - along with her mother, father, and brother - survived the blast, although her grandmother did not. Sadako became a healthy, active girl, a champion runner for her school class. But radiation can wreak its deadly effects for many years after its release, and Sadako was diagnosed with leukemia when she was 11. She spent many months in the hospital. While she was in the hospital, Sadako began folding paper cranes because, according to a Japanese story, a wish will come true if you fold a thousand paper cranes. Sadako very much wished to be well again, so she determinedly folded more than 1,000 cranes. But her wish did not come true and she died in 1955 at age 12. However, some of her cranes were used to line her casket; others were given to her friends. Her classmates decided to create a memorial to Sadako, wishing to remember their friend and her courage and kindness. A collection was begun, culminating in a statue in the Peace Memorial Park in Hiroshima, which was unveiled in 1958. At the top of the memorial is a figure of a girl holding up a large crane. Thus, folded paper-cranes came to symbolize the desire for peace in the world. At the foot of the statue are these words: