This should work, your problem is that you included a #
in your id, this is not needed here.
This would work.
$('.friend_chooser_pop').html("<div id='friend_chooser_list'></div>");
$('#friend_chooser_list').html('something');
If this is your intend to include #
in the ID and you want it to work you can use it like this:
$("#\\#friend_chooser_list").html('something');
It's escaping the #
and allowing jQuery to get the right element anyway. I would not recommend this, it can get confusing pretty fast.