You can add a customized Google search engine to your site. It can be customized to display your own logo and colours, and to return customized search results. Google offers a paid version, a free ad-supported version, and a free ad-free version for non-profit organizations and educational institutions. To see samples, read documentation, and to setup a search engine, see Google’s page.
How do I add a search engine to my site?
Google Custom Search Engine