This is not uncommon. Coughs get worse at night. Do you think this is due to a cold/flu type of situation or something like GERD which can have a night time cough. Gerd is acid reflux. If you think it is post nasal drip from his cold, a very common reason to cough more at night, propping his head up a bit more when sleeping can help. Adding pillows, towels under his pillow, etc. Does he use a humidifier? He could try some tea with honey. There is always cough medicine. But I am not a huge fan of cough medicine because coughs are supposed to move gunk through our lungs. If we don't cough, it sits. Festers. Possibly goes from being a cold to something like bronchitis which is bacterial. No shortness of breath, right? https://www.medicalnewstoday.com/articles/319498#home-remedies
Cough gets worse at night & sometimes it happens with me. The whole day I am okay & before going to bed I start coughing. Try gargling with warm salt water before going to bed.