The importance of the necklace relates to the way in which Debbie relates it to possibilities and potential outcomes of her life. The very first chapter shows us Debbie playing with her necklace and how she desires her life to be like it, so she could open spontaneously and let new experiences pass through:
Debbie picked up her necklace and jiggled the catch. It stuck sometimes in a partly open position, and the connecting loop could slip out.
The way in which the necklace is first lost by her and then how it passes through the hands of a number of the main characters clearly consolidates the theme of being open to new experiences and also how the lives of the characters "criss cross" with each other during the course of the novel. It is, however, Hector in Chapter 38 who returns the necklace to Debbie, indicating the way in which they almost but not quite start a relationship.