The changes I'd make here are as follows:

-separate the markup from the javascript somewhat more by
taking in an element to the autocompleting function.

-separate the data handling from the display.  This would make it pretty easy to build an autocompleter that queries the network for stuff to match.

-separate the rendering.  I don't know what you want to do here.  My initial impression is to append a div as the next child of the parent of the targeted input, and put stuff in there.