Dear A.:
My son had some on his fingers. We got a good wart remover from Kroger (any department store will do). We used DuoFilm, a Salicylic wart remover. Other brands have a similar ingredient and probably work well, too. You cannot use it for eyelids, genital areas and lips, but eyebrows should be OK, because the skin is neither thin, wet or in a fold there.
Expect a little sting, one to two weeks of ongoing treatment and some discoloration to brown where you put it on (goes away after the treatment). You can get skin colored band-aids to put on both for the looks and to make sure it does not rub off too soon. Wash your hands after contact and don't inhale that stuff, just to be sure.
It did not show any difference for a week, then it healed up. Don't give up too early. It's probably best if you put it on while he lies down for a few minutes to prevent it from going into his eyes.
Alternatively, a doctor may offer a one-time treatment, which 'freezes' the wart. It may take a while to go away, but you don't have to remember to put stuff on twice every day.
Good luck. In addition to the not-so-great looks, warts are dangerous in that they can spread. So close to the eyes I would definitely do something, not wait it out. I am NOT a doctor, so you should double-check at least with the pharmacist if not his pediatrician.
BTW, if he wears glasses, you may want to put a band-aid or something on the frame so that it does not rub against it. Even if he looks dorky for a while, I would play it safe.
Regards,
W.