I figured it out by looking at another plugin that uses a class.
It looks like I needed to pass the object by reference.
I changed these 2 lines:
register_sidebar_widget(‘mytestClass’, array(‘mytestClass’, ‘widget’));
register_widget_control(‘mytestClass’, array(‘mytestClass’, ‘control’));
TO
register_sidebar_widget(‘GoogleWeather’, array(&$this, ‘widget’));
register_widget_control(‘GoogleWeather’, array(&$this, ‘control’));
