There is actually a website called Book of Mormon Games. Surprisingly (not really) the url is http://bookofmormongames.com/. What there is on the site is a list of short multiple choice quizzes about different "topics" from the Book of Mormon. For example, there is a quiz about the different "Villians and Victories" in the Book of Mormon (here). Additionally there is a Book of Mormon Hangman game. A word is displayed, well, a few letters and spaces is displayed. You guess letters (you can only get it wrong 4 times before you lose the game). There isn't any actual hangman displayed, just a you got it right or you got it wrong kind of presented text.
One problem I found with the site is that on the quizzes you answer all the questions and hit the "Grade Me!" button and nothing happens. Even after the time limit counts down to the end, still nothing happens. Must be some problem in the coding or scripting of the quizzes.
All in all, the site could help you with your Book of Mormon knowledge a little bit, if the quizzes were scripted correctly. I'm just not sure I'd call it a website about games (although the hangman game would constitute a game).