Haila and Nick
These two characters run the eateries in the archipelago. Nick runs the Diner on Verdure Island, and Haila manages the Cafe on Sprout Island. Neither one of the owners will ever leave their businesses, so there isn't much you can do with them interaction-wise.
Nick and Haila are important when it comes to getting cooking recipes. If you give them cooking ingredients, in exchange they will give you recipes for your kitchen. They can each give you one recipe per day, and you can get multiple individual recipes from the same cooking ingredient.
For example, you can give an Egg to Nick to get the recipe for Egg Over Rice, then give another Egg to Haila for the Boiled Egg recipe. The next day, give Nick another Egg for Mayonnaise, and Haila an Egg for Spa-Boiled Egg. When they are out of recipes to give you for a specific ingredient, they will let you know ("I can't think of anything to make with this ingredient").
To get their Sun Stones, you need to learn a lot of recipes from them. For Nick, you will need to learn 40 cooking recipes, and Haila requires 30 cooking recipes. You don't have to actually make these dishes in your kitchen; just learn the recipes from them. When you learn your 40th (for Nick) and 30th (from Haila), you will be given the Sun Stone.