Ьыл незакрытый td
</td>
'new_area_name' без кавычек были
<script src="//ajax.googleapis.com/ajax/libs/jquery/1.7.1/jquery.min.js"></script>
<div class='area'>
<input type='hidden' class='area_id' value='{$area_id}'>
<select class='area_edit'></select>
<button class='area_edit ok_btn' text='OK'>OK</button>
<label id='area_name'>{$area_name}</label>
</div>
<script>
$(".ok_btn").click( function(){var parent = $(this).parent();
console.log('parent='+parent);
if ( parent.hasClass("area"))
{console.log("ok_btn clicked")
;
var new_area_name = parent.find(".area_edit option:selected").text();
var new_area_id = parent.find(".area_edit option:selected").val();
console.log(new_area_name);
parent.find(".area_id").val( 'new_area_id' );
parent.find("#area_name").text( 'new_area_name' );
parent.find("#area_name").show();
parent.find(".area_edit").each(function(){$(this).hide()})
}
})
</script>