I didn’t even know that they made an Elf on the Shelf game until I found this today!
The game comes with a little elf that plays music. One player hides him, and the first of the other players to find it gets a present in their sleigh. The first player to get three presents wins. It sounds simple enough for toddlers, yet fun enough for the whole family, and would make a great accompaniment to your adopted Elf on the Shelf!
(Parents with an interesting sense of humor might want to check out The Elf off the Shelf: A Christmas Tradition Gone Bad by Horace the Elf. 😉 )