The Mighty Basketball is the replacement for the Mighty Eagle in the Angry Birds Seasons episode, "Ham Dunk". It works the same way as the Mighty Eagle does in this game.
It could be bought by real money but was removed in newer updates. After being bought, it would also unlock the Western and Eastern Ham Dunk levels.
It worked the same way as Mighty Eagle, as said above, but instead of launching a sardine, the player had to launch a backpack that would summon the Spalding Ball.