You could use google analytics event tracking for this. Attach it to the search form submit, or the search results page, and pass the search keywords to the google analytics event tracking call. If you are OK with this method, you need to do two things:
1. Find out how to use google analytics event tracking
2. Find out how to hook up the event tracking code to the search feature in orchard *.
* I actually haven't seen or used the search feature so I can't guide you on how to do this. It's probably as simple as overriding a .cshtml file in your theme or module so that you can add the GA tracking code.