RSS Widget: Pulling in a random feed URL

I have a client who wants to pull in a number of different RSS feeds into the standard RSS Widget that comes with WordPress. So basically, upon each page refresh, it would look at an array of feed URL’s and select one to display its’ feed. Then on the next refresh, a different feed.

I haven’t found a plugin that can do this, does anyone know of a way to do this or some plugin that might do what I want? Nothing too crazy. I like the way the RSS widget works, just want the ability to pick the RSS feed each time.