The example you've provided will work if you're using a case insensitive collation such as utf8_general_ci. (See the MySQL Character Set Support manual section for more information.)
Alternatively, you could simply use the LOWER function as follows:
$names = strtolower(mysql_real_escape_string($_GET['part']));
$result = mysql_query("SELECT * FROM namestable WHERE names LIKE LOWER('%$names%') LIMIT 10");
Incidentally, if you're attempting to catch differences beyond simple case changes, you could also use MySQL's SOUNDEX function to obtain and find a match for strings that sound similar.
Incidentally, you need to use mysql_real_escape_string on your $names variable, or (better still) use prepared statements via the mysqli or PDO interfaces.