I'm not sure if this answers your question, but I imagine you're looking for a search form in addition to a button.

I use Zoom Search, a $100 piece of software from a company called Wrensoft (if your site is smaller, you may be able to get away with the free trial) that spiders and troubleshoots my site. It also indexes pdfs and other file formats. I know they sell a DW plugin version of the product, but I use the stand alone app.

I know Google has a search tool for free for webmasters, but last I looked - the results looked like Google... not the parent site.

hope that helps