I`d always assumed that the Google searchbox on someone`s site would direct to Google`s own page when a keyword was entered. It was only when I was looking around a site that I saw that when I entered a word to be searched on the web the list of URLs was displayed on one of that site`s pages. Is that how it`s normally done? If so I think it`s a great idea since it tends to keep the user on the site that little bit longer. I have to admit that I normally use the toolbar at the top of my browser to do any searches, so using the searchbox on a site itself is new to me.
That sounds like google custom search http://googlecustomsearch.blogspot.com/ http://www.google.com/coop/cse/ http://www.doshdosh.com/how-to-set-up-google-custom-search-and-make-money/
Search results all really depend on the feeds there our using there has been cases were you type in car on a web site brings up adult listings. But for the most part there pretty effective. and make really good money for website owners. If you have good quality traffic i could certainly recommend how to move forward.