Jquery clone elements: problems retaining checked radio buttons

First of all, to uncheck the new radio input you should use

$(this).prop("checked",false);

Second, your original radio input is getting unchecked because at the moment of cloning it, the new element has the same name and id of the original one, and you change it after cloning, which doesn't help.

A way to avoid this would be simply saving the original radio and resetting it after the cloning and name changing is done, like so:

$('.add_btn').click(function(){
    // clone the previous element (a "repeatable" element), and insert it before the "add more" button

    // save the original checked radio button
    var original = $('.repeatable').last().find(':checked');
    $(this).prev('.repeatable').clone().insertBefore(this).html();
    // get the number of repeatable elements on the page
    var num = $('.repeatable').length;
    // again, get the previous element (a "repeatable" element), and change the header to reflect it's new index
    $(this).prev('.repeatable').children('h2').html('Person ' + num);
    // now, go through all text boxes within the last "repeatable" element...
    $('.repeatable').last().find('input').each(function(){
      // ...change their "structure" data attributes to reflect the index+1 value of the "repeatable" element
      dattr = $(this).data('structure') + num;
      $(this).attr({
        'id':dattr,
        'name':dattr
      // update the "for" attribute on the parent element (label)
      }).parent('label').attr('for',dattr);
      // clear the input field contents of the new "repeatable"
      // if the type of the input is "radio"...
      if ($(this).attr('type') == 'radio') {
        // remove the checked attribute
        $(this).prop('checked',false);
      // for all other inputs...
      } else {
        // clear the value...
        $(this).val('');
      }
      // check if there's a checked radio button, and restore it
      if (original.length == 1) {
          original.prop('checked',true);
      }
    });

Working example: http://codepen.io/anon/pen/ByKEYJ?editors=101

I also added values to the radio buttons.