[j0hnik]

<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'), x;
    (function show(){
    	 do index = Math.floor(Math.random() * li.length);
    	 while(x == index);
         li.eq(x).removeClass('show');
    	 x = index;
       li.eq(index).addClass('show');
       setTimeout(show, 1000);
    }());
</script>

без повторений