Link's Awakening is my all-time favorite Zelda game. I'd love to see a remake. I wouldn't change the story too much or extend it too much or anything though. I did think the same thing as you when I played Link Between Worlds, that Link's Awakening would look great in this style.
I loved the original's island setting. It set a certain mood. Also, rudiementary as it was, I also liked some of the text. It was almost poetic. "Earth, Wind, Sea, Sky, All on the lid of a dreamer's eye." Some of the crytic text even explains what happens to the island when you wake up: "Somday thou may recall this island in the waking world. That memory must be the dream world." That and the secret ending you can get in the GBC version explains things a bit, I think.
But yeah I love that game. All the great things about Link to the Past in a more compact form. Great for replaying, too. Not too short and not too long.


