I resolved this issue myself and figured I would post my method.
In zigweather/widgets.php I disabled this:
<img src=”<?php echo $zigweather->get_iconurl($info)?>” alt=”” width=”120″ />
And I then replaced it with:
if(($zigweather->get_description($info)) == Clear) {
echo “<img src=’your-own-custom-image-url’ width=’120′ />”;
} else if(($zigweather->get_description($info)) == Sunny) {
echo “<img src=’your-own-custom-image-url’ width=’120′ />”;
} else……….etc
Obviously “your-own-custom-image-url” should be the url to the image you want displayed. Good luck with that!