But if you won't change to doing it in PHP (you really should) at least do this:
First, replace the existing <script>...</script> that creates the banner with just this code:
<a href="http://padrewa.com"><img id="mybanner" style="height: 120px; width: 900px; border: none;" alt="Banner"/></a>
That's right. No scripting there, at all. Just HTML.
And then, at the *bottom* of your page, JUST before the </body> tag, put in this code:
var pix = [
"http://farm3.staticflickr.com/" + pix[Math.floor(Math.random()*pix.length)];
Yes, I tested it. Yes, it works.