Fairly early on in Pokemon Legends: Arceus you'll get a quest - number 66 - called The Sea's Legend from Professor Laventon himself. It's the most difficult to solve in the entire game, because it refers to a book called The Sea's Legend that you can't actually read in this game. It's actually a book that you can find in the recently released Pokemon Brilliant Diamond and Shining Pearl, which is just fine and dandy - unless you don't have that game or haven't found/remembered that book.
But thankfully, internet sleuths have figured it all out. Huge kudos and thanks to HenryHeliolisk on Reddit for figuring all of this out! (opens in new tab)
Here's the multi-stage solution to solving the mystery of The Sea's Legend in Pokemon Legends: Arceus.
1. Catch Buizel, Mantyke, and Quilfish/Overqwil
First, you're going to need to catch three specific Pokemon in order to begin to solve the Sea's Legend. You'll need Buizel, Mantyke, and Overqwil. Thankfully all three are found in the Cobalt Coastlands where the Sea's Legend mystery can be solved.
Buizel can be found in the Aipon Hill and Bathers' Lagoon areas.
Mantyke is located in the Ginkgo Landing and Tranquility Cove areas.
Overqwil can be evolved from a Qwilfish (found in Tranquility Cove, Lunker's Lair, and Islepsy Shore) by using Barb Barrage in Strong Style 20 times.
2. Have the three Pokemon in your party and head to the Coastlands Camp
Once you've caught all three of those, make sure you add all of them to your party. From there, head to the Coastlands Camp and rest until the evening.
3. Head for the giant stone arch
When it's night, look just to the right from the Coastlands Camp across the water and you'll see a huge stone arch that's not named on the map. Swim over there on Basculegion and go right through it.
If you've done it right, you'll hear a noise and a text box will pop up that reads - "You heard something that sounded like a cry and a boulder sitting in the distance".
4. Make your way to the Seaside Hollow
Although it doesn't exactly match what the textbox says, you'll then need to head to Seaside Hollow, which around the other side of the Cobalt Coastlands' point. Head inside the cave.
5. Battle and catch the Phione and Manaphy
Inside you'll find three Phione and one Manaphy that you can battle and catch. They've each got a little protective shield around them, so you'll need to toss a Pokeball first and then a Pokemon to start the battle process (or a Pokemon twice if you don't want a waste a ball).
And that's it! The mystery of the Sea's Legend solved! All that's left to do is go and talk to Professor Laventon and tick the pesky quest off the list.
This is the only way to catch both Phione and Manaphy so definitely worth all the effort.