Google has introduced a new sitelink feature where it will show a site search box above the sitelinks for a website in the search results.
The new change, in addition to the search box getting a prominent spot just under the site description, also takes advantage of auto-complete when people type something into the search box.
The new search box is available for both desktop and mobile searches. Here it is for desktop:
Here is the mobile version of the new search box.
Google provides documentation so webmasters can enable this so any searches done within the sitelinks search box will direct people right to the results page on your site. You need to have a working search engine on your site, however...