Location: The Hideaway (ship)
Petitioner: Jote
Unlocked: Complete Through the Maelstrom
Reward: 150 Magicked Ash, 1 Meteorite, 30 Renown
In order to unlock this side quest you need to complete Through the Maelstrom, a main story quest that will take you out to sea. Return to the Hideaway and Phoenix, Heal Thyself will be waiting. The petitioner, none other than Jote, is waiting in the Infirmary.
Speak to Jote and you can ask her about the side quest. She is worried about Joshua, whose condition continues to worsen, and she wants to try something stronger to heal him - or at least mitigate Joshua's symptoms. She wants Clive's help, and if you agree she'll ask you to track down some Stonawart. Apparently there is some in Rosaria, in aether-flooded areas that are unsafe for normal humans. (Fortunately, Clive is not normal.)
Jump to the Three Reeds Obelisk and travel west through Three Reeds until you come to a bridge. Beyond it is the Trans-Rosarian Aqueduct. Leave the main road behind and take a left, into the marsh. Follow the path and you'll eventually come to some ruined housed, populated only by an Akashic Gigas and some Akashic Weavers. If this lot are still giving you trouble this deep into the game, then... well. Needless to say, take out the Weavers first.
With the Goblins gone you can poke around. Check the south side of the clearing to find three samples of Stonawort. Take them back to Jote to earn your reward and end the quest.