This quest is unlocked after completing the main quest "Marooned". To start it, you must visit your manor in Great Inagua and read a letter that you will find on a coffee table in one of the rooms.
Edward receives a letter from an old friend, Stede Bonnet.
Captain Stede Bonnet
Caribbean Sea
Mystery Island
Caribbean Sea
Stede Bonnet's Portrait
Caribbean Sea
The Last Voyage of Stede Bonnet
Caribbean Sea
Mystery Island is located in the southeastern corner of the map. Head there and find the schooner grounded off the island's west coast. That's Bonnet's ship you're looking for.
Anchor your ship in the designated area, jump into the water, swim to Bonnet's ship, and climb aboard. Check the hatch leading below deck. The hatch will be locked, and a group of pirates from Bonnet's crew will appear.
Approach the campfire where the pirates are waiting and talk to them. You will learn about a second group of pirates who have mutinied and stolen the key to the hatch.
Follow the pirates as they take you to the other side of the island, where you meet the mutineers. The encounter with them will begin with a conversation, but will soon turn into a fight.
It is possible to attack the mutinous sailors before the dialogue ends, gaining a small advantage in the process.
The weather will change drastically, it will start raining, and you just have to help kill all the rebels.
As soon as the fight with the mutinous sailors ends, a British ship will appear near the island, and a few soldiers from it will swim ashore.
Get rid of them. You can use a small gunpowder keg lying on the beach where the soldiers come ashore, for example.
Follow the loyalists back to Bonnet's ship. This will start a short cutscene in which Sharp will hand Edward the key to the hold, and Edward will go below deck.
Progress through the hull until you reach a spot to interact. Doing so lets you enter Bonnet's cabin.
To complete this quest objective, simply approach the main table and read the letter from Bonnet, which is pinned to it with your old pal's dagger. Once you do this, a cutscene will play, after which you will automatically leave the ship, taking Stede Bonnet's portrait with you.
However, before you do that, take a look at the other items in the cabin: the sugar crates that Bonnet lost at the beginning of your acquaintance and apparently managed to recover, a fragment of his journal lying on the round table,...
... or the Act of Grace lying on the small desk next to Stede Bonnet's portrait. Doing this will earn you additional information.
Hint:
Return to the manor at Great Inagua and head toward the basement. The right place is in the hallway between the stairs and the treasure room.
And that is the end of this quest. As a reward for completing it, you will receive a new sea shanty.