The snow-on-the-mountain camellia flower is given to Jem by Mrs. Dubose. The flower could mean several things, but at first the reader thinks she is giving it to him to remind him that he ruined her flowers. This is not what she meant to do. She didn't want him to feel guilt.
However, after helping her get through her fits of withdrawal by reading to her each day, Jem was a distraction for her and a blessing. She wanted Jem to know that she appreciated his help. Even though it was "punishment," he did his duty each day and helped her through some major fits. She gives that flower to him as either a "thank you" or possibly even a "forgive me" symbol. She wanted him to know that she wasn't angry or holding a grudge when she was ready to die. It was a sweet gesture, and Jem is mature enough to understand--as he is seen holding the flower and caressing its petals later on in his room.
Interestingly, the camellia is the state flower of Alabama, the state where Maycomb is located. Having been imported from Asia, the red camillia is symbolic of longevity and the white of faithfulness. Mrs. Dubose's camellias are apparently white since in Chapter 11, she calls her camellias the "Snow-on-the-Mountain." Therefore, they are symbolic of longevity, and they do survive Jem's brutal hacking with Scout's baton. In order to demonstrate that they have prevailed, Mrs. Dubose sends Jem a perfect Snow-on-the-Mountain in a candy box. When Jem opens it, he becomes angry, but Atticus tells him,
"I think that was her way of telling you--everything's all right now, Jem, everything's all right...."
Although Jem flings the box into the fireplace, he "fingers" the wide petals thoughtfully. This white camellia is a peace offering, an offering for a long, long time.