Ian at Conversation Marketing has an interesting tutorial about integrating Google Analytics into your Google custom search engine.
In the past I have always tracked internal search data using a simple sql database and trawled through the queries every so often to check that users were able to navigate the site without resorting to search. If you see that users are always searching for certain pages or products its a good idea to add these to your primary navigation.
Using Google Analytics to trawl through this data is a much better idea, even if you use a non-Google search engine for your site. Just filter the results in your Analytics account by title or url and you should be able to see some valuable data from your sites search engine straight away.