My 14 year old son keeps getting painful vein/ headache on one side his head in the afternoon, what could be causing it?

Numerous things. Headaches need proper evaluation by a physician. Hypoglycemia can cause them; unilateral throbbing headaches can also be migraines; being misaligned can cause headaches. Medications can cause headaches as well. He needs a thorough history and physical to understand the nature of them, try and determine cause and then offer appropriate treatment.
Migraine headache. Migraine headache is unilateral or one sided usually preceded by aura. Mother is having same problem. Make sure it's not eye strain or visual problem, if with vomiting it could be serious., .