Zelda Wiki:Sandbox

The Mermaid's Necklace is an item in the Link's Awakening Trading Sequence. It is obtained by presenting the Fishing Hook to the Fisherman found underneath the bridge at Martha's Bay. The Fisherman, who lost his hook when a fish took it, will gratefully accept the Fishing Hook. In return, the Fisherman promises to give Link his next catch, which happens to be the Necklace.

The Necklace can be returned to its rightful owner, Martha the Mermaid, who will then allow Link to take a Scale from her tail.

Trivia
In the Japanese and select European versions of the game, Link receives a bikini top instead of a necklace. This explains the Mermaid's shy and anxious behaviour around Link. The bikini top, most likely deemed too inappropriate, was changed to the Necklace in the American release.