Forbidden Sea by Sheila A. Nielson
It seems that Adrianne Keynnman's life cannot get any worse. Her father died years ago and his fortune is gone, so the Keynnmans are now poor. Adrianne's Aunt Minnah hates her and her mother is afraid to stand up to her. One night, she finds her sister, Cecily stranded on a rock on the beach. As Adrianne goes out to rescue her, she sees the Windwaithe mermaid trying to take her sister into the sea. She stops the mermaid from taking Cecily, but discovers later that the mermaid is really after her. Adrianne soon realises that the mermaid's call is hard to resist. Will she give in and leave her mother and sister who need her?