Chronic non-productive cough could be due to different reasons but general possible causes include an allergic reaction, lung or respiratory tract diseases, few disorders or probably due to any malignancy.
Since the cough is non-productive, the possible suggestions include a kind of asthma, postnasal drip, gastroesophageal reflux disorder (GERD), chronic obstructive pulmonary disease most likely from smoking, or presence of any tumor. Not to forget to mention the possibility of tuberculosis which sometime could have no other feasible symptoms.
We cannot tell if the cyst he got is associated with his chronic coughing until an examination will be done. As cough itself is not a disease but a sign that there possibly could be a disease, going to have a check up is the best option. Investigation like X-ray will somehow give the idea of the cause of the chronic cough. There are also other tests that will be done possibly to find e proper diagnosis.
The chronic cough can be just like an ordinary cough but it could also mean something more serious. Taking cough remedies without knowing the cause won't be a good idea. My best advise is to see your doctor to have your son examined and do the needed investigation for an appropriate diagnosis. 6 months coughing is quite a long period.
Take care and have a Happy New Year!