Choose a class when adding a text widget

On my user’s site, there will be about 4 text widgets. These should change rather frequently, (old ones deleted, new ones created) and will be heavily edited by the (novice) user.

I would like to create 2 or 3 styles for the widgets in the stylesheet, and have the user be able to choose which class will be attributed to the widget when they save it.

I understand that I could view the source of each widget after it is created and style the unique ID it in the stylesheet, but this is time intensive and clunky since it requires either the client to tell me when they’ve added a new widget, or me to notice when they have.

I’ve done some searching and I am really surprised that there is no TextWidget plugin that adds a “choose class” dropdown to the widgets. Is it because it is impossible? How might I go about this if I chose to hack something together?