Levels of Distractibility: Who cares? = *. Some people will be distracted from your meaning: = **. Many people will be distracted from your meaning: = ***.
Question: How can I break my son of the habit of putting "me" in the subject of the sentence?
Answer: Don't lecture him. Simply repeat his statement, substituting the "He" and "I." "He and I went to the game." It'll take a while, but he will gradually gain the habit of using "he," "she" and "I" in the subjects of his sentences.
By the way, students will probably only use this expression in compound (two) subjects. You won't hear any student say "Me am going to the game."
Rating of Distractibility: ***. Your son will either break himself of the habit or he will be mightily embarrassed when he finds himself in a situation in which he wants to speak correctly. Students might have the "right to their own language," but listeners also have the right to label people who use "him" and "me" in the subject of their sentences as "illiterate."