I have used the following in the past:
http://forresst.github.io/2012/06/22/Make-a-list-jQuery-Mobile-sortable-by-drag-and-drop/
Here's a JSFiddle of it in action using something as simple as:
$("#sortable").sortable({
items: "li:not(.ui-li-divider)"
});
$("#sortable").sortable();
$("#sortable").disableSelection();
$("#sortable").bind("sortstop", function (event, ui) {
$('#sortable').listview('refresh');
});