A Qiqirn and former employee of Buscarron's Druthers forgot some of his personal effects upon leaving. Go speak to Baensyng on Hawker's Alley in Limsa Lominsa to learn where you might find said Qiqirn.
You learn from Baensyng that a Qiqirn merchant on Hawkers' Alley named Kyokyoroon may know something of Teteroon's whereabouts. Bring the trader a chicken egg to coax him into spilling the goods.
After receiving the chicken egg, Kyokyoroon is more than happy to tell you that Teteroon has since moved on to barter his wares at Memeroon's Trading Post in upper La Noscea. Proceed to the outpost and present Teteroon with his forgotten scarlet earring.
You have delivered the scarlet earring to Teteroon. Reminded of Buscarron, he promptly lapses into nostalgic reminiscences of former days.