On the island, cut down the grass near the big rock in the middle of the island to reveal a small tunnel. Crawl through it and jump into the hole to go into a mini-dungeon.
This dungeon is usually filled with Chuchus, but, due to limited supplies, you cannot reach the main parts of the dungeon. You can fight a few Red and Green Chuchus to get their Jellies, which can restore hearts and magic, respectively.
Outside of the island, before you leave, look around for a submarine-thing on the sea. Cruise/Sail to it and talk to the people on-board. You’ll get a Treasure Chart for your trouble.
After this, continue sailing east to Dragon Roost Island .