Hello and hope you are doing well.
His symptoms could be due to an infected tooth. Infected tooth roots from the upper teeth can go inside the maxillary sinus and cause sinusitis. This can cause the headaches. He might need a CT of the sinuses for more information. However, if he does have an infected tooth, he will still need to have this addressed as dental infections can be potentially life-threatening.
Hope this helped and do keep us posted.
Get a flash light (or other light source) and look carefully at the tooth that has the ache problem. If you don't see anything obvious it may still be a good idea to see a dentist.. well maybe a general practice doctor would be a better place to start.
You seem to say the symptoms come and go during the day, this in itself is an unusual symptom in my experience. I've had bad tooth pain in my life, but while in the early going it may have been stimulated by eating something cold, when it got to really hurting, it just hurt all the time.