In Link's Awakening, the Necklace is the twelfth item in the game's Trading Quest. It is obtained from the Fisherman beneath the bridge in Martha's Bay, after Link gives him the Fishing Hook. After giving him the Hook, the Fisherman promises to give Link his next catch, which happens to be the Necklace. The Necklace actually belongs to Martha the Mermaid, who had lost the item to the waves while swimming. In return for her Necklace, she lets Link take one of her Scales.
↑"Simply put, she originally lost her top in the Japanese version, which is why she looks and acts so flustered. This is also why she's so quick to flee when Link tries to take a peek while diving." — Unused Parts, Artemis251's Link's Awakening Cache.