Remove below line, it's removing the dialog from DOM on close.
beforeClose: function(){ $(this).remove(); }
Continued...
$( '#videodialog' ).dialog({
autoOpen: false,
height: 380,
width: '50%',
position:[285,140],
modal:true,
resizable: false,
draggable: false,
open: function() { $("#video").attr('src','http://www.youtube.com/embed/b16V25eNyJY'); }
beforeClose: function(){ $("#video").attr('src',''); }
});