I'm not aware of a means to review a history/report of what users have searched for using the DNN Search (yet, though I'm sure that will be coming at some point).
You can switch DNN Search for other DNN-based search modules that will enable history/reports.
Open-SearchEngine
Here is one advanced search module (a really nice one) and the required add-on component to enable reporting of user searches:
Module: http://www.snowcovered.com/Snowcovered2/Default.aspx?tabid=242&PackageID=5800
Reporting add-on: http://www.snowcovered.com/snowcovered2/Default.aspx?tabid=242&PackageID=6044
The Open-SearchEngine (OSE) module and its reporting add-on represent a very serious upgrade to the stock DNN search. Besides page content, the OSE module can index the content of linked documents such as Word docs and PDF files. You can set the module to cover more than one site and the site(s) do NOT have to be DNN-based. This can be a real boost for admins of multiple, related websites. Further, each site can be configured to be indexed using a separate set of security credentials (so your DNN site could be configured to index secure pages on a separate SharePoint intranet site, for instance). In short, the module rocks.
You can easily add the OSE search as a skin object (simply swap it your skin code with the existing DNN Search skin object). Results are reported within DNN and are configurable. The module itself makes no changes to the core so you can easily swap back to DNN Search or some other search if desired. There are more notable features -- see the links above for details.
Free DNN Search Options
So, as usual, the obvious free option is Google site search. Use it in combination with a free "Google Webmaster Tools" account in order to get user search reports for your custom site search (note: this is different than Google Analytics reports that tell you what terms users used to GET to your site but not what terms they search USING your local site search).
There are pros and cons to using an external search engine for a DNN site. If you want to play with it, though, I recommend trying the free Google AJAX custom site search with in-page results (i.e., the results show up within your DNN page instead of in a new page). Creating your own custom site search allows you a good bit of flexibility in what site(s) to search, what types of content to include/exclude and allows you to influence search results by tweaking things on the backend. I like it and use it so I blogged about it: you can see a working demo, get example code for adding to your DNN skin and read brief explanations of the search options here: http://googleAJAXsearch.com
Good luck!
-mamlin