The temptation to cheat while grading his own paper is very great. I think that this is a wonderful story of moral development in children. Far from being very worried about what would cause your nine year-old son to cheat, I think that your heart should be soaring because your son has internalized a conscience and a sense of honesty that compelled him to confess his cheating to you. Your parenting and modeling of honesty as a cherished value has been taken to heart by your son. He has shown courage and has brought honor to himself and should be praised accordingly.
Kids cheat all the time when they correct their own papers.
Teachers know this and factor that temptation into this practice. There is nothing wrong with letting kids correct their own papers and see how and why they made certain mistakes. If the self-correcting is done in coordination with explaining how students should have arrived at the correct answer, then it's fine. In other words, using the correcting of tests as an extension of the learning process is a good learning technique. If a teacher merely hands tests back to kids and reads off the correct answers, there is no real learning going on.