I have updated your jsfiddle - http://jsfiddle.net/bUHVc/6/
You were missing the include for jquery. Also, you did not need the find("img") line in your code. You can easily accomplish what you are looking for using the animate() function.
The jQuery
$(".arrow").hover(function(){
$(this).stop().animate({"opacity": "1"});
}, function(){
$(this).stop().animate({"opacity": "0.5"});
});
The CSS:
.arrow{
opacity: 0.5;
}
The HTML:
<script src="//ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<a href="#" id="right-button"><img class="arrow" src="http://placekitten.com/200/200" alt="right" /></a>