Shaky hands can be due to stress and anxiety, too much consumption of caffeine or alcohol, low blood sugar (hypoglycemia), autoimmune disease like multiple sclerosis, thyroid disorders, hypertparathyroidism, low vitamin B12, nerve compression in lower cervical or upper thoracic veretbrae, and liver disease.
Since you also have similar symptoms, hyperthyroidism, hyperparathyroidism, liver disease, vitamin B12 deficiency, and autoimmune conditions should be ruled out. Also, since he is a teenager, you must check on his caffeine intake (tea, coffee, cola etc). Take care!
The medical advice given should not be considered a substitute for medical care provided by a doctor who can examine you. The advice may not be completely correct for you as the doctor cannot examine you and does not know your complete medical history. Hence this reply to your post should only be considered as a guiding line and you must consult your doctor at the earliest for your medical problem.