We only hear Judy's name once, when Christopher first arrives in London. Otherwise, we mainly know her as "Mother." For the first half of the book, we think she's dead, since that's what Christopher has been told by his father. In reality, she ran away with the next-door neighbor, Mr. Shears.
So we don't really know much about her until Christopher discovers the letters she's been writing him every week for two years, but that his father has been hiding from him. In the letters, she describes her new life in London and apologizes for leaving him, but also provides some intense self-analysis. She bluntly says, "I was not a very good mother," describing herself as impatient, and admitting she often was really angry with her son. Quite the confession.
When she and Christopher reunite, she's overjoyed to see him again (although horrified that her ex-husband told their son she was dead this whole time). But, despite her best efforts, once tasked with caring for Christopher again, she all-too-quickly shows the same traits she lamented in her letters.
It's through Judy that we get a glimpse of just how challenging it is to care for someone with a social disorder, like when Christopher starts smashing things in a store, or refuses to step onto a bus and insists that they walk miles home. We admire Judy's awareness of her limitations, and her attempts at being a better mother, despite all the difficulties she faces and the craziness spinning around her.