ureech, так надо?
<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 max =$('#hide_right li').length;
setInterval(function () {
var min = 1;
randomLi = Math.floor(Math.random() * (max - min +1)) + min;
$('#hide_right li.show').removeClass('show');
$('#hide_right li').eq(randomLi-1).addClass('show');
}, 1000);
</script>