Save jQuery UI Sortable on WordPress

if you use the stop function when you intialise .sortable() you can ajax send the new order to a script, which (when you write it!) would save the new order to the database.

jQuery(function(){

jQuery("#sortable").sortable({

    stop: function (event, ui) {

        var new_order = jQuery(this).sortable('serialize');
        jQuery.post( "http://www.thissite.com/wp-admin/ajax.php", { action: my_custom_ajax_save, order: new_order }, function( data ) {

            console.log('ajax sent and response received');     

        });
    }
});

})

If you give more context we could help with the actual save. For example, if its a post you could save to the postmeta database nicely, what exactly are you saving?