I agree it could well be sinus 's it may also be allergies ..let us know how he does ..
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.
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 .