chosen with bootstrap 3 not working properly

Applying Chosen after the modal is shown should solve your problem:

$('#myModal').on('shown.bs.modal', function () {
  $('.chosen-select', this).chosen();
});

Or when Chosen was already applied, destroy and reapply:

$('#myModal').on('shown.bs.modal', function () {
  $('.chosen-select', this).chosen('destroy').chosen();
});

Fiddle here: http://jsfiddle.net/koenpunt/W6dZV/

So in your case it would probably something like:

$('.someClick').click(function(e){
   e.preventDefault();
   x.modal('show');
   x.on('shown.bs.modal', function(){
       x.find('.modal-body').load('path/page.html', function(response, status, xhr){
           if(status == "success"){
               $("select[name=elementName]").chosen();
           }
       });
   });
});

EDIT

To complement Chosen you can use the Chosen Bootstrap theme