My friend has one enlarged pupil accompanied by sharp headaches, but not at the same time. what could this be?

Worth checking: Difference in pupil size, known as anisocoria, requires prompt consult with a physician as it may indicate some brain problems. I would refrain from naming diseases that can cause aniscoria to spare unnecessary fear, but i would advise him to get to the doctor promptly.

See an eye doctor: There are several conditions that could lead to mismatched pupil sizes, a common one being migraine. The timing of the headaches and pupil changes is important, as is your friend's medical history and general health. Pupils should normally be about the same size, so any difference should be evaluated by an eye doctor.