What can cause the side of my right eye to hurt when I blink?

Corneal irriation. Or conjunctival irritation due to a localized abrasion. If you are a contact lens wearer, you must look for infection. I would recommend getting it looked at if not better spontaneously within a day.
VariousPossibilities. If discomfort occurs with each blink, it is likely that there is some obstruction or impediment to the movement of the lid, either on the lid or eye surface. Start with lubricating artificial tear drops and if not improving see your eye doctor.