How can Waverly's character be evaluated in terms of behavior, quotes, actions, and changes?
Another approach to an essay on Waverly would ask the questions "what is invisible strength, why does her mother think it is important, how does (or does) Waverly acquire it, and what does it look like when / if she does?" The story opens with a statement about it, and concludes with an image of it. Your thesis would be something like "In learning the meaning of "invisible strength" from her mother, Waverly develops a stronger sense of self." Another approach would be to consider Waverly's bicultural identity. In what ways is she Chinese? In what ways is she American? How do these different cultural identities conflict with each other in the story? How does or does she resolve them through playing chess?
You could discuss Waverly's complex feelings toward her mother. At first, she sees her as a support, but as the story progresses the mother-daughter relationship is strained. Discuss the increasing pressure Waverly feels as her chess ability grows, and her mother's expectations for Waverly. Discuss the end of the story, when Waverly and her mother have a huge fight, and how the relationship changes from it.