Finding Princess Ruto is part of the "Spiritual Stone of Water" main mission in The Legend of Zelda: Ocarina of Time remake. The Princess of the Zora Domain is missing, and her father, King Zora asks Link for help locating her. To complete this quest and progress through the campaign, Link must play detective and collect a few pieces of information to discover where he needs to go, which involves obtaining the Silver Scale to reach Lake Hylia.
If you want some help with this quest, here's where to find Princess Ruto in Zelda Ocarina of Time. Once you talk to King Zora in the southeastern side of Zora Domain, he will tell you that his daughter is missing and he has searched for her as far as Lake Hylia — the exact place you must go. You just need to figure out how to get there.
By talking to a Zora near the entrance of the Zora Domain, you will learn about an underwater tunnel that connects the domain to Lake Hylia. Although Link can swim and dive, the tunnel is too deep underwater for him to reach. To allow Link to dive deeper, you must obtain the Silver Scale.
Head back to the room where you met King Zora, and go northwest to find a Zora at the top of the waterfall. Talk to them to participate in a diving training, in which you must collect five scales underwater. After completing the challenge, they will give you the Silver Scale.
Jump off the ledge and dive into the lake. Now that you have the Silver Scale, you can reach the tunnel to the west. After swimming through the tunnel, you will emerge at Lake Hylia.
Near the Zora swimming in front of the entrance to the area, you can see an object emitting a blueish glow underwater. Dive to collect a bottle with a scroll inside. Return to the Zora Domain and, once you're out of the water, open the bottle to read the paper inside.
Extract — continue reading at the source.