My original paper has already been published in the conference proceedings. Now, the body organizing the conference has invited me to submit an extended and revised version of my paper to be included in a new book.
In my experience, extended versions most typically contain three classes of new information:
- More extensive prose and examples to explain, discuss, and otherwise clarify the material presented in the original paper. Conference papers are sometimes rather haiku'ed due to page limits.
- Important details omitted from the original paper due to length. For example, a discussion of the key points of an algorithm may become a full presentation of its details, and a proof sketch may become a proof.
- Extension work, either enhancing evaluation or extending the set of material studied, as suggested by the original paper's future work, feedback from the reviewers, questions raised in the conference talk, etc.
Some combination of these is almost certainly the right sort of material for your extended paper, but which combination exactly is something that only you, your collaborators, and your advisors are in a good position to determine.
I think the easiest way to understand the possibilities for what you might do is to look for examples of papers in your field that have been through the same process. If you can find some then you use these tangible examples as templates for making your own improvements.
You might also benefit from asking some people to read the paper, and then asking them which parts they were confused and interested by. For example, you might find that the section on one aspect of your theory is actually a little bit to complicated for non-experts, and that one aspect of your findings seems to have interesting implications for a specific area of research. Once you know these things you can expand these sections to create a better, but also longer paper.