Brace yourself: the cure that Pliny suggests involves urine.

While I was trawling through the relevant secondary sources (and I am very glad for Google Books, as digitized previews have been of immense use so many times) I came across the individual of Salpe, a Greek midwife or female healer who is cited by Pliny the Elder several times in his Natural History as an authority on minor remedies. Modern commentators have varying opinions on whether Pliny trusts Salpe, or whether Salpe can be identified with another figure of that name who wrote some erotic stories, but that is irrelevant.

On the subject of sunburn, Pliny relates:

"Salpe recommends that the eyes should be fomented with [urine], as a means of strengthening the sight; and that it should be used as a liniment for sun scorches, in combination with white of egg, that of the ostrich being the most effectual, the application being kept on for a couple of hours."

This is taken from the Natural History, Book 28, Chapter 18. The translated text can be found online in the Perseus Digital Library!