Filter Text from sidebar widget?

In the method “widget” look for the line that sets the category then add either:

$cat = empty($instance['cat']) ? 'N/A' : $instance['cat'];

(where you replace N/A with something you like) or… better… add a filter instead like this:

$cat = apply_filters( 'widget_my_cat', empty($instance['cat']) ? 'N/A' : $instance['cat'], $instance, $this->id_base);

And then hook into ‘widget_my_cat’ to set it to something else if the value is ‘N/A’