Let's say common cards do 1 extra damage against uncommon cards, uncommon cards do 1 extra damage against rare cards and rare do 1 more damage against common cards.
What do you think?
You have to remember that on lackey, people can just pick and take whatever cards they want, regardless of how uncommon it is to find them in "packs," so this would essentially be the same thing as giving them an additional type, which would honestly be a lot easier as it'd most likely be printed on the card. Unless you have some way to very easily identify the rarity of a card, such as printing it as suggested above, it would kinda work. However there's still that problem with the fact that card rarity has nothing to do with obtaining them in lackey, so I'd honestly say adding an additional type is a much better bet.
Yeah, I'd rather see the mechanic done through more of a Small > Medium > Large flavor. Really its just rock/paper/scissors with a million different names