A new online Book of Mormon game called, appropriately, The Book of Mormon Game has recently been released (here). The premise is that a passage from the Book of Mormon is read out loud and it is your job to identify where the passage is located in the Book of Mormon (book and chapter), as quickly as you can.
Eric Poulin is the man behind the game. He mentions playing the game as a missionary. From the About section of the site:
While serving our missions in San Fernando, California, we'd play this game during interviews with our President, as we waited for our turn.
The first time I played this as a new missionary, I was amazed at how fast the other missionaries seemed to be able to find the right place. I love this game because it inspired me to study and learn the scriptures and become like those other missionaries. Now we present it to you. We hope it will also inspire you to become very familiar with this great book.
Eric Poulin (Ontario, Canada)
Shane Hartley (Washington, USA)
You can log in (via the site or using your Facebook account) and your times will be recorded. He has also created a Facebook app (here).
I think that this idea does have the potential of getting people to become more familiar with the scriptures, as was Eric's goal with putting it out. He's also working on a New Testament version (here), but says it needs a little more work before it is completely ready to go.
Lastly, Eric shared with me another game he's put together (it was qualified with a "needs graphic work and some further development"). It's called Don't Forget the Lyrics! and can be found here.