A central theme is the moral or lesson that the author makes about a subject through the story. This means a theme is the deeper meaning of the subject discussed.
One of the dictionary definitions of idea is "an attitude of mind; opinion; belief; impression." This definition allows for the central idea to be a synonym of the central theme, which would make an idea the deeper meaning of a story.
Idea is also a synonym for motif. In reference to literature, motif is a term that refers to a unifying element in a work. For example, a recurring character type, subject, image or symbol. This definition of idea looks at a central idea as being different from a central theme.
The Bedford Glossary of Critical and Literary Terms gives the example: "The subject of a work might be suffering. The theme, depending on the view of the individual author, might be that suffering is in God's plan and should therefore be accepted." In this example, the idea of the work is also the subject: suffering. The theme is the statement the author makes about suffering.