Prerequisites: Attain level 2 - 3 friendship with Derrek and Mariele
Reward: Two Moon Cloves
Unlocks: --
It's difficult to say what, exactly, unlocks this quest. I had it trigger upon visiting the Police Department after 15:00 on a Saturday. I also had my friendship with both Derrek and Mariele at the third level, and both of them were involved in the cut scene that followed. So. Given that she's more important to the overall story, it's possible you only need to befriend Mariele.
Once the scene is triggered you'll see Mariele fretting over her lost sewing gear, which was last seen on a bench to the north of the Lazy Bowl Tavern. Once the scene ends you'll be directed to check in at Silky Stitch Tailor's, between 10:00 and 18:00. (It's only open from 11:00 to 17:00, every day but Saturday, so you may need to change your plans a bit.)
Hannah, the shop's assistant, has Mariele's Unfinished Scarf, and she hands it off to you. Take the scarf to Mariele. She likes to wander town a lot, though she seems to spend a fair amount of time around Town Hall, as well as at her own home in the southwest of Moonbury. Hand off the missing item to complete the quest.