PHP Notice – Custom Function

$page = get_page_by_title($title) – this line is failing somewhere, so you should do a check on this to make sure it exists.

Like so:

function the_alt_title($title="") {
    $page = get_page_by_title($title);
    if (!$page) {
        return $title;
    }

    if ($p = get_post_meta($page->ID, "_x_entry_alternate_index_title", true)) {
        $title = $p;
    }

    return $title;
}
add_filter('the_title', 'the_alt_title', 10, 1);