Using Add from Server to upload by post ID

You can try to add this into your functions.php file:

add_filter( 'upload_dir','wpse87234_upload_dir');
function wpse87234_upload_dir($uploads){

    // check if we are in the "Add From Server" media tab
    if (isset($_REQUEST['post_id']) && isset($_GET['tab']) && $_GET['tab']=="server") {
        $prefix="post_";
        $post_id=(int)$_REQUEST['post_id'];

        // change the uploads info to our needs:
        $uploads['subdir']="https://wordpress.stackexchange.com/".$prefix.$post_id;
        $uploads['path']=$uploads['basedir'].$uploads['subdir'];
        $uploads['url']=$uploads['baseurl'].$uploads['subdir'];
    }
    return $uploads;    
}

This should give directory like /wp-content/uploads/post_1234/ with post_id 1234.