I always used to look at the search button at various websites and wonder , will it not take the users to various other search engine sites and thus losing the stickiness of the site. Now I think there’s a better way of handling the search box on websites. Here’s a product from Yahoo labs which addresses that gap.
Y!Q Beta :

Embed Y!Q into your site to give your users convenient, related searches

    * Deliver a more engaging experience for your users
    * Keep users on your site with search results overlaid directly on your web pages
    * Easily embed Y!Q in 5 simple steps