You will need to use Javascript for this because PHP is processed after you submit the form or if you use AJAX then that's a different and inefficient story.
Javascript
<input type="text" value="" onKeyPress="this.value = this.value.toLowerCase();" onKeyUp="this.value = this.value.toLowerCase();" onKeyDown="this.value = this.value.toLowerCase();">
Server-side PHP
$_POST['s'] = strtolower($_POST['s']);
// Do whatever you want with the search term now because it is lower-case