Is there a JS event triggered when a new widget is created?
It appears when a new widget is created it has a variety of classes (such as new and loading-new) that don't exist on established widgets:

class="widget new loading-new myplugin_mywidget flipped last"


It's also possible to add javascript to the widget settings file that gets rendered for your widgets settings. In the admittedly lame (but illustrative) example below, your new widget will get the text color of red, and after it is saved and the page reloads, no red color will be applied.

{% includejs %}
(function(){
    $('.widget.new.myplugin_mywidget').css('color', 'red');
})();
{% endincludejs %}