window.addEvent('domready',function(){
    
    var input = $('filter');

    // set the title attribute of every element
    // to it's text in lowercase
    $$('ul > li').each(function(item){
        item.set('title', item.get('text').toLowerCase());
    });
    
    // the function we'll call when the user types
    var filterList = function(){
        var value = input.value.toLowerCase();
        $$('li').setStyle('display','none');
        
        // check the title attribute if it contains whatever the user is typing
        $$('ul > li[title*=' + value + ']').setStyle('display','');
    };
    
    // make it happen
    input.addEvent('keyup', filterList);
    
});

<p><input id="filter" type="text" /></p>
<ul>
    <li>Sonic the Hedgehog</li>
    <li>Knuckles the Enchinada</li>
    <li>Tails</li>
    <li>Amy Rose</li>
    <li>Dr. Robotnik</li>
    <li>Rouge</li>
    <li>Shadow the Hedgehog</li>
</ul>

jsFiddle