I would have to say Link's Oasis (well, that's after you get the deed for only a few pendants) would have to be the most relaxing place because I've rarely seen it rain in that quadrant of the Great Sea for starters. In addition, the climate seems perfect, for it isn't within the area of Dragon Roost or the Fire and Ice islands. There's a waterfall on the island with many lawn chairs and sandy beaches, accompanied by beautiful palm trees, creating a relaxing scenery already.
The interior is quite nice, too, with a giant, roaring fire for when light turns to night, and the spirit of the butler will assist you whenever you're down. You could also give the King of Red Lions a rest by using one of the small boats on the island, although I suspect those can be used in the interior lagoon. I'm not certain there's another place as serene as the Oasis.