Location: The Hideaway (ship)
Petitioner: Nazaire, Cursebreaker
Unlocked: Begin Back to Their Origin
Reward: The Breath of Lightning (Pile Drive), 40 Renown, 800 gil
In order to unlock this side quest you need to progress through the majority of Final Fantasy XVI. Once you've fought Ultima during the Streets of Madness main story quest you'll be returned to the Hideaway, and you'll need to speak to several important NPCs. Once you're done talking with Jill a bunch of new side quests will pop up, including Trial and Error.
The petitioner, Nazaire, is standing at the bottom of the stairs leading up to the Mess, on the Boarding Deck. Nazaire has a new recruit who wants to become a Cursebreaker, specifically a scout, and he wants Clive to overlook the boy's evaluation. Nazaire will direct you to Northreach, where the trial will take place.
Fast Travel to Northreach and you'll find Nazaire and the recruit, Ember, standing a short walk away to the east, just beyond the gate to Claireview. Nazaire will send Ember off to steal a log book from the imperials, as a test of his mettle. Follow Ember out of Claireview, east to the Dragon's Aery, and then up the northern hills of the Dragon's Aery to the guard tower of Eastwatch.
Things aren't going well when you arrive, and Clive will find Ember being menaced by a Tot Aevis. You'll need to kill the beast. The Tot Aevis is just like every other Aevis you've ever fought, and shouldn't prove a big deal. You'll receive a Dragon Talon for winning the battle, and you can check near the bottom of the tower for a chest containing a Kestrel's Beast Braid.
Travel back to Northreach and speak to Nazaire, who's standing where you left him. Ember will stroll up, and after a quick conversation you'll have to return to the Hideaway. Speak to Nazaire - he's back where you started the quest - and you can choose whether or not to recommend Ember as a scout. The choice you make ultimately doesn't matter to your tangible reward, so go with your gut. The quest ends here either way.