How to change icon for a widget tab?

You can change an icon for a widget tab by applying the CSS rule. 
For example, if you have added new tab with name "favorites", then you can add the following code:
.uploadcare-dialog-tab-favorites:before {
  background-image: url(http://your_sprite.png) !important;
  background-position: 0 50px;
}
.uploadcare-dialog-selected-tab.uploadcare-dialog-tab-favorites:before {
  background-position: 0 0;
}
	

http://your_sprite.png should be 100×200 image with active and inactive states.

The same way you can customize icons for standard tabs.