How can I make a module's configuration exportable by features? At a high level, what needs to be undertaken to take a module's configuration data and make it exportable by Features?

Whether this is a contrib module that does not yet have Features support or a custom module that I have written, where can I start so that I can get it in code?
Instructions are in the Features API.txt, which provides several examples of how to better integrate your module with Features.

The easiest way seems to be building your module around the CTools Export API. Here's a pretty good article explaining how to do that.