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

The piece of glass Arthur gives to Gerda comes to represent faith and hope. She carried it with her for a very long time because he'd asked her to remember his words, "Look at that pillar. It is safe. We have to have faith. Never forget it, Gerda." 

Gerda later lost the glass, but she never forgot Arthur's words.