You could stick "sponsored" tweets into the twitter feed.
Moreover the Twitter API TOS seem to forbid the notion of other sponsorship and advertising for this widget unless Twitter also gets a cut:
> "In cases where Twitter content is the primary basis of the advertising sale, we require you to compensate us [...]"
It sure sounds like this widget is all about Twitter content.
edit: looking below, I guess there are restrictions against this. if not the tweet feed, maybe in some other area of the widget.