The Castle in the Sea — Scott O'Dell

When Lucinda's wealthy and powerful father dies, she inherits the Isla del Oro, a small island off the coast of California, where everything is as it was when her grandfather first discovered it a century before. Lucinda is now the wealthiest girl in the world -- but her guardian forbids her to leave the island, and strange things begin to happen to Lucinda, until she begins to think someone is trying to kill her. But who? Her mysterious guardian? Her handsome fiance? Or the island doctor?