The life of the son in Cormac McCarthy's novel The Road is one in which a post-nuclear world is described. The mother left after giving birth to her son. The son, at one point, tells his father that he wishes to join his mother and his father tells him never to say what he did again.
In a letter from the mother, to her son, one would need to decide why she thought she had to leave. Perhaps she could not find the courage to live in a world like the one they are going to be forced to live in. Perhaps she could not face the fact that she will be required to raise a child in such a bleak place. Perhaps she simply fails to relate to her son and feels that his life would be better without her.
Once this is decided, try to take on the persona of the mother. Explain what happened during the son's birth which led her to the decision. Be sure to acknowledge her feelings about the situation and why she felt she had to leave. Detail how her leaving would benefit the life of the son.
An example of a letter would look something like this:
Dear son, I am sure that you cannot understand why I left. I do not want you to think that this is your fault and I feel the need to explain why I left you and your father to face the world without me. I simply could not face the fact that I brought you into this world to face such a desolate and hard life. I felt like I was the reason you would suffer in life and could not look into your accusing eyes every day. I had no choice but to leave you with your father.