Why does Gerda keep the piece of glass that Arthur gives her for so long? What does it come to signify to her?

The piece of glass given to Gerda by Arthur is a symbol of faith and hope. She carried it with her because Arthur asked her to remember his words, "Look at that pillar. It is safe. We have to have faith. Never forget it, Gerda."

Eventually, Gerda lost the glass, but she never forgot the words.