I would advise saving only the information you need, rather than copying the entire div in one go. This would allow you to cherry-pick the data your really need and safely persist it across page loads.
The best solution for this would be to use local storage. This allows you to save string-to-string values persistently across a site as far back as IE8 (see more information here: http://www.html5rocks.com/en/features/storage).
For example, you could save your data like this:
localStorage["carouselSource"] = JSON.stringify(
$('#carouselDiv li img').map(function(i,e){
return $(this).attr('src');
}).get();
);
and retrieve it like this (before the jCarousel call):
var items = jQuery.parseJSON(localStorage["carouselSource"]);
jQuery.each(items, function(i,e){
$('#carouselDiv').append('<li><img src="' + e + '" width="65" height="65" alt="" /></li>');
});
This would ensure that the same set of images are displayed when the user returns to the page. You could use a similar technique to save things like the position of the slider and any other variable data you want to save.