Good question!

(Hat tip to Adam Whitehead of the Atlas of Ice and Fire)
The ordinal names of the continents – Westeros, Essos, Sotheryos – are a bit odd, because in order for them to be named that way, “Westeros” has to be thought of “west” of something, Essos has to be thought of “east” of something, and Sothoryos has to be thought “south” of something.
Since “os” and “rys” and “ros” are used in Valyrian, my guess is that these names emerged from Valyria. This makes sense from an ordinal perspective: Westeros was the strange and forbidding land to the west of their empire, and Sotheryos is immediately south of the Valyrian peninsula. The odd case is that of Essos – why not call your home continent something akin to the “Middle Kingdom,” think of it as the center of the world?
My guess is that the term was originally applied to the lands “east” of the Valyrian peninsula, given that the Valyrians’ oldest foreign relations were with the empire of Old Ghis, and then expanded outwards from there to include the whole landmass as Valyria’s knowledge of the world expanded and especially when their empire grew to encompass much of it.

