He may have bronchitis
which can be caused by a virus or bacteria, in the absence of a fever
it is more likely a viral bronchitis. Bronchitis simply means inflammation or infection of the bronchi. It usuallly presents with a wet cough which may be persistant and worse at night.
Your son may benefit from the use of a mucolytic
to break up the mucus, a bronchodilatory medication (some of the coughing in bronchitis is because the airways are closing, if they are reopened then the coughing may decrease- mucolytics and bronchodilators are often combined into one cough syrup. Some children , if the above do not work and the airways are significantly inflammed, may need a short 3 day course of oral steroids
to reduce the inflammation.
If he has a runny nose or nasal congestion
, the mucus may be running back into his throat and contributing to his coughing when he lays down, you can use a decongestant (either oral or topical) to treat any nasal symptoms.
I hope this helps