What books, other than The Curious Incident of the Dog in the Night-Time, deal with abandonment of a child because of the child's illness? I would like to find some books which represent one of the parents leaving because he or she can't cope with the family situation. I would like to support my argument that the mother leaving in The Curious Incident of the Dog in the Night-Time is justifiable.

Just to clarify, the reasons for Christopher's mother having left, as indicated in her letters, are that she was unhappy and lonely in her marriage and believed that her son was better off with his father, who knew how to calm Christopher down. Christopher's mother ultimately loves her son. Concerning the topic of child abandonment, I would not recommend using the term justifiable. Instead, you can discuss how parents having left their children eventually led to something beneficial, rather than attempt to argue that child abandonment is right or reasonable.
The Memory Keeper's Daughter by Kim Edwards (2006): Dr. David Henry delivers his own twins; his son is perfectly healthy, while his daughter has Down's Syndrome. Wishing to protect his wife, Norah, he tells the nurse, Caroline, to take the baby to an institution in secret. Instead, Caroline moves to a different city and raises the daughter, Phoebe. The child grows up to become a lively and happy woman, loved fiercely by Caroline.
Follow the Stars Home by Luanne Rice (2001): Dianne Robbins and Tim McIntosh learn that their infant daughter has Rett's syndrome, and Tim flees. Dianne rises to the challenge of single parenthood, while also receiving help from Alan, Tim's brother. You could reason that the child is better off without Tim, who would not have been a supportive father. Alan demonstrates the caring abilities of fatherhood, and eventually, Dianne and Alan begin a romantic relationship.
The Secret Garden by Frances Hodgson Burnett (1911): This is an old novel but a classic for many reasons. Archibald Craven is ashamed of his sickly son, Colin, and hates that the boy is a constant reminder of his wife, who died shortly after Colin was born. Archibald barely acknowledges his son. Colin finds a friend in Mary, an orphaned girl his age. The secret garden comes to life with the memory of Colin's mother, healing both the children. Eventually, Archibald recognizes that Colin needs him, through a letter and a dream about his parental responsibilities.