There are a couple of reasons why Jing-Mei's mother wants Jing-Mei to be a prodigy. First of all, Jing-Mei's mother came to the US to escape the hardships of her former life in China and to start over. As such, she views the US as a place where dreams can come true. In the opening paragraph, for example, Jing-Mei says that her mother believed you could be anything you wanted to be in the US:
You could become rich. You could become instantly famous.
Secondly, her mother is influenced by Auntie Lindo, whose daughter is a chess prodigy. Lindo talks, for instance, about how her daughter plays chess all day and wins lots of trophies. This constant bragging creates a sense of competition within Jing-Mei's mother. Moreover, Auntie Lindo's daughter is proof that you can be anything you want to be in America. Together, these ideas become highly influential for Jing-Mei's mother.
Jing-mei’s mother wants her to be a prodigy because her friend Lindo Jong’s daughter Waverly is a chess prodigy. She also wants her daughter to be successful.
Jing-mei’s “Aunt” Lindo has a daughter named Waverly who is a chess prodigy. Jing-mei’s mother decides that her daughter should be some kind of prodigy too, because she is tired of Waverly’s mother bragging about her success.
I looked out over the audience, at my mother's blank face, my father's yawn, Auntie Lindo's stiff-lipped smile, Waverly's sulky expression
Waverly has become famous. Jing-mei has a sort of rivalry with her, because she does not enjoy being compared to her.
Jing-mei’s mother also sees a variety of other prodigies in magazines and on television, so she thinks her daughter should be a prodigy too.
Jing-mei’s mother sees America as a land of opportunity, where anyone can be a success.
You could open a restaurant. You could work for the government and get good retirement. You could buy a house with almost no money down. You could become rich. You could become instantly famous.
Jing-mei’s mother lost everything in China. She lost her family, and her twin babies. Her daughter is her last hope. She wants Jing-mei to have it all. She wants her to be a success because she never had the chance herself.