Does he fall asleep on his back if he does get him to sleep on his side , this could be sleep apnea or some sinus problem , maybe good to ask the doctor .
I would need more information. Does he cough while awake? Has he had a recent cold? Does he have allergies? Does he sleep on his back? Have you looked at his throat?
I would say that if he is normally healthy, that it may be sinus drainage even if he doesn't have a runny nose or cough while awake. It could take some time for the drainage to pool and cause the coughing. Taking out the tonsils won't help with this either. ( I am a chronic allergy sufferer and this is what my doctor told me as a child). Unfortunately, doctors seem reluctant to test and diagnose allergies in young children. My poor son had to wait until he was 12 before I could give him some real help even though he suffered from age 2 and up. This is very frustrating as a parent and pediatric allergy medicine does not work.
Try having your son drink some water before bed and sleep with his head slightly elevated to see if that helps.
I agree it could well be sinus 's it may also be allergies ..let us know how he does ..