Man, we have already been burned by too many poor adaptations of video games to other media. There has yet to be a worthwhile movie made, and most of the Anime that I have seen has been an abomination. Anyone remember the Anime based off of Final Fantasy 5? It was a solid adaptation if you left out the characters, changed the setting, changed the plot, and gave it a really, really creepy S&M feel.
If we are going strictly Anime, most Western made games would be out. Anything by Bioware, Bethesda, or the other Western studios would not translate well. Even if we broadened the scope to allow western animation or live action most every game has had an equivalent show that is better. Mass Effect? I will just watch Star Wars or Firefly. Dragon Age or Elder Scrolls? I can watch Game of Thrones or The Lord of the Rings. Fallout? Road Warrior. Even going back to some of the older games you have the same problem. If you ignore the pretty bad adaptation of Wing Commander, it would not work in the film or television mediums because Battlestar Galactica did everything it could hope to do but better and first. The one thought I had was an over the top satirical adaptation of Grand Theft Auto, but then I remembered they basically did that in Canada and called it the Trailer Park Boys.
That leaves us with Japanese games. Metal Gear seems obvious, but the later games were one giant cut-scene anyway. I would be loath to see another Final Fantasy adaptation; if they screwed up the earlier games (which I know Square would, or would allow someone else to do) I would never forgive them and the latter works are not worth the time. No More Heroes would be fun for about 5 episodes, but it lends itself more towards a short run. I might watch that, but if they expect to milk it for multiple seasons they would ruin it. This is the problem with most games.
Really the best choice I can see is the Phoenix Wright games as a long-running episodic show. It would be Japanese Law and Order for kids/teens. You can fit a case easily into a half hour format and the art style already lends itself to the medium. It's over the top and goofy while still having just enough grit to pull in viewers over the age of 15.