Apparently the phrase "hairy eyeball" has a variety of meanings in colloquial senses, but this is a real one. It is a "limbal dermoid" - a type of teratoma that happened to develop on a young man's cornea, as described by physicians in Iran for the New England Journal of Medicine:
A 19-year-old man presented to our ophthalmology clinic with a mass in his right eye that had been present since birth but had gradually increased in size. He did not have pain, but the mass caused vision defects, mild discomfort on blinking, and the intermittent sensation of the presence of a foreign body. Physical examination revealed a white, ovoid mass, 5 mm by 6 mm, that straddled the inferotemporal limbus (Panel A), with several black hairs (Panel B). Visual acuity was 20/20 in the left eye and 20/60 in the right eye. Intraocular pressure was normal... The lesion was excised, and lamellar keratoplasty was performed for cosmetic reasons. Pathological findings confirmed the diagnosis of limbal dermoid. As expected, there was little improvement in visual acuity after surgery because of the amblyopia and induced astigmatism.