[рони]

ureech,

<script src="https://code.jquery.com/jquery-3.3.1.min.js"></script>
<style>
    .show{display: block;}
    li {display: none;}
</style>
<ul id="hide_right">
    <li>Item1</li>
    <li>Item2</li>
    <li>Item3</li>
    <li>Item4</li>
    <li>Item5</li>
</ul>
<script>
    var li = $('#hide_right li').get(), el = rnd(li), temp = [];
    function rnd(li)
    {
       var index = Math.floor(Math.random() * li.length);
       return li.splice(index,1)
    }

    function show()
    {
       li.length || (li = temp, temp = []);
       $(el).removeClass('show');
       temp = temp.concat(el);
       el = rnd(li);
       $(el).addClass('show');
       window.setTimeout(show, 1000)
    }
    show()

</script>