Manual:$wgAjaxSearch

When true, this enables auto-suggestion for the search bar, using asynchronous javascript XML requests.

This also requires $wgUseAjax to be true.

Details
Setting $wgAjaxSearch = true; causes wfSajaxSearch to be added to $wgAjaxExportList

mediawiki/skins/common/ajaxsearch.js is no doubt related in some way (how?)