Customizing the contents of the Page Not Found page in any WordPress theme can be tricky, and usually involes editing the 404.php template file. The Genesis Framework already abstracts the default content of the page into a function, and this plugin can unhook that and replace it with your own custom title and content.
Use [genesis-404-search] shortcode to add a search form to the page.
If you’d like to dynamically list content (ex: most recent posts), I recommend you install the Display Posts Shortcode and use it in the 404 page’s content.