Wish I had found this thing a while ago when I was developing my own auto-complete functionality. This extremely small jQuery plug-in (about 7 Kb) looks and works brilliantly. The plug-in takes advantage of jQuery more than other plug-ins that do the same thing, is styled ONLY with CSS (and uses no images) and is very responsive. It creates all the necessary elements dynamically and only needs a regular INPUT box.
It is a very customisable plug-in, the data can be supplied to it in either an object on the page or loaded in JSON format from a URL. And of course, it’s free.