At one point prior to Link's arrival on Koholint Island, Martha modeled for the alligator artist, Schule Donavitch, living in Animal Village. The mermaid statue was named The Mourning Mermaid and became the artist's "masterpiece". It was also placed on the shore of Martha's Bay.
Also prior to the events of Link's Awakening, Martha apparently lost her Necklace. If Link finds it for her, she will allow him to take a scale from her tail. Link places this scale on The Mourning Mermaid, revealing a hidden passageway beneath. Inside the passageway, he finds the Magnifying Lens.
In the original Japanese release as well as a few foreign releases, Martha had actually lost her swimsuit top rather than her necklace. However, foreign Nintendo localization teams felt that this was too explicit and changed it to the necklace in the American release.
As part of the Link's Awakening DLC, Tetra has a Martha Recolor of her Standard Outfit.