Is it possible to view search queries
However, Google Analytics keeps track of searches done via your site. You could just set up Google Analytics and simply copy / paste the tracking code into the footer area in the dashboard.
One idea is to override the Search block or make a copy of it with a new name, then add logging to it.
There is a do_search() method in the block controller that gets the query string. You could retrieve the query string, sanitize it, and save it to a specific database table. This could be done by adding another table in the Search block's db.xml file.
To make the log easier to access, you could make a dashboard single page that reads the logging table and lists the results.
Cool idea though to make the feature though - Another project for my 'spare time'! :p
I'll take a look at the block code later MrK - sounds like a lot of the legwork is already done.