I am writing a thesis statement about the relationship between the child and his mother in Oryx and Crake and how her leaving the child affects him...

With Margaret Atwood's book Oryx and Crake, there are many ways you could push this thesis statement further. First you need to determine exactly what you want to argue, and remember—thesis statements must always make a debatable point that you can argue, and ideally, should convince the reader that your point has validity based on the examples you provide from the text to support it. 

You are correct in saying that the main character, Jimmy, is abandoned by his mother, and this event affects him in many ways, including psychologically. However, this is not necessarily a debatable point because almost anyone would agree that a person would be affected by a parent's abandonment in childhood. What you need to do to "push it further" is explain how this event affected specific aspects of Jimmy's adult life, and why that is significant.

One example of this could be Jimmy's lack of attachment to women throughout his adult life. He has constant meaningless encounters with women and immediately pushes them away when they try to get close to him or get to know him better. This can be connected to the abandonment of his mother, and one could argue that her abandonment is what causes him to detach himself from women as an adult from fear of being abandoned by them as well. 

Another example is Jimmy's interest in bending the rules of society with his friend Crake (Glenn). Because Jimmy's father is very busy with work and his new romantic partner, after his mother abandons him, Jimmy is not supervised very often. He participates in many questionable activities with Crake such as exploring pornographic websites, watching live executions, and engaging in some computer hacking and minor theft. His mother leaving not only provides better opportunities for these activities to take place but seems to lead to Jimmy's desire to bend the rules and go along with Crake's behaviors—regardless of what they are.

As an adult, Jimmy is later employed by Crake and unknowingly participates in the destruction of the entire human race. These activities are foreshadowed by his early activities with Crake, especially because they strengthened the bond between these two characters. Jimmy also suspects his mother of participating in illegal activities of protest from the little bit of communication he has with her throughout the book. This could be an explanation for why he later supports Crake's desire to change the world and improve the human species.

By highlighting these later events and how they can be connected to the abandonment of Jimmy by his mother, you can strengthen your argument, as you are commenting on the significance of this traumatic event. Make sure to use specific examples such as this, and always connect them to your original point—that Jimmy is greatly affected by his mother leaving him as a child.