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.