Suggestion: It appears that the searches sometimes stacked up in queue if I typed fast. (i.e. if I typed a long sentence quick the results continued to flash different content many times, playing catch-up. Looked through the JS quick and I see you are trying to prevent this with runningRequest1/2 so maybe it's the short delay in the .fadeTo() after the GET callback. Try moving the runningRequest = false into the fadeTo callback.
Also, .fadeTo() has a default queue of it's own. You could also add .stop() like this:
$('#function').html(data).stop().fadeTo('fast',1);
Otherwise, really cool! I use "site:php.net foobar" most of the time but this would be a great little widget.