It appears Westeros is the name given to the land by the Westerosi themselves. I have yet to find where the name came from though west for Westeros, east for Essos and south for Sothoroyos does seem plausible.
However, note that is is not called Westeros to everyone, the Dothraki call is "Rhaesh Andahli" and Viserys calls it "Our land".
Somewhere beyond the sunset, across the narrow sea, lay a land of green hills and flowered plains and great rushing rivers, where towers of dark stone rose amidst magnificent blue-grey mountains, and armored knights rode to battle beneath the banners of their lords. The Dothraki called that land Rhaesh Andahli, the land of the Andals. In the Free Cities, they talked of Westeros and the Sunset Kingdoms. Her brother had a simpler name. "Our land," he called it. The words were like a prayer with him. If he said them enough, the gods were sure to hear. "Ours by blood right, taken from us by treachery, but ours still, ours forever. You do not steal from the dragon, oh, no. The dragon remembers."
A Game of Thrones, Daenerys I
From the above it appears that those in Essos, at least those in the Free Cities, also refer to Westeros as Westeros too.
The Children of the Forest do not call it Westeros though mention that men do, perhaps meaning the name could have come from the First Men.
"Gone down into the earth," she answered. "Into the stones, into the trees. Before the First Men came all this land that you call Westeros was home to us, yet even in those days we were few. The gods gave us long lives but not great numbers, lest we overrun the world as deer will overrun a wood where there are no wolves to hunt them.
A Dance with Dragons, Bran III
However, from a later quote by Samwell it appears as though even if the First Men called the land Westeros there is no written memory of it.
"The Others." Sam licked his lips. "They are mentioned in the annals, though not as often as I would have thought. The annals I've found and looked at, that is. There's more I haven't found, I know. Some of the older books are falling to pieces. The pages crumble when I try and turn them. And the really old books . . . either they have crumbled all away or they are buried somewhere that I haven't looked yet or . . . well, it could be that there are no such books, and never were. The oldest histories we have were written after the Andals came to Westeros. The First Men only left us runes on rocks, so everything we think we know about the Age of Heroes and the Dawn Age and the Long Night comes from accounts set down by septons thousands of years later. There are archmaesters at the Citadel who question all of it. Those old histories are full of kings who reigned for hundreds of years, and knights riding around a thousand years before there were knights. You know the tales, Brandon the Builder, Symeon Star-Eyes, Night's King . . . we say that you're the nine hundred and ninety-eighth Lord Commander of the Night's Watch, but the oldest list I've found shows six hundred seventy-four commanders, which suggests that it was written during . . ."
A Feast for Crows, Samwell I
The Andals came to Westeros from Essos so it could easily mean "West of us" as you say.
For thousands of years the Andals abided in Andalos, growing in number. In the oldest of the holy books, The Seven-Pointed Star, it is said that the Seven themselves walked among their people in the hills of Andalos, and it was they who crowned Hugor of the Hill and promised him and his descendants great kingdoms in a foreign land. This is what the septons and septas teach as the reason why the Andals left Essos and struck west to Westeros, but the history that the Citadel has uncovered over the centuries may provide a better reason.
The World of Ice and Fire, Ancient History: The Arrival of the Andals
As for an out of universe reason, or Geroge's intention, on what Westeros means I haven't found anything. I will update if I ever do.