For a grammar game to actually be engaging, it would have to move beyond a simple right or wrong binary. Instead, the fail sequence would have to be entertaining enough that we keep trying. It would have to have a feedback mechanism that lets you know if you have a near miss.
Parts of speech should be weapons. Just as Batman has a variety of gadgets in Arkham City, and you have to learn the correct circumstances and combinations to use them, so too, should you have adjectives, adverbs, and verb tenses in your “utility belt.” Better yet, there is something wrong in a paragraph, and you have to aim the correct word at the flaw to correct it. Angry Birds, but with Words.
Overall, this would mean changing our framework from thinking of grammar as a norm (social, linguistic, majoritarian, etc.) you conform to. Instead, gamers would experience grammar as a tool that has uses, and increases their power to act. Instead of right or wrong (binary), grammar in a game world has more effective and less effective (including utterly ineffective) uses.