Funny you should bring this up, since I just wrote up a post on my blog about this topic, after having walked through several other forum threads that had posted this question. I'll give the thumbnail sketch to what I discovered and my own feelings:
- I was surprised at how popular 3rd person was. Most games are 1st person it seems, and yet there is definitely a desire for 3rd.
- One of the criticisms regarding 1st is the sub-genre of POV. Some people think that POV games are really attempts by the dev to avoid having to render yet another character. Perhaps. There is also the problem that with strict POV, some of the "fun stuff" is going to happen where you can't see it. And oh yeah, when your character gets a kiss from someone in the POV game, assuming your eyes are open, you should be seeing a lot of forehead. Instead you often see puckered lips, meaning you're getting kissed right on your open eye. Weird.
- Narrative issues. Diegesis is a style of storytelling in which the viewer is told what is happening. This can be through direct narration or narrative-like techniques such as choosing a camera angle or even showing a scene somewhere else -- planet of Endor, Palpatine's throne room, bridge of the Rebel ship, back to Endor. A 1st person story with the MC as narrator would be autodiegetic. That can be limiting since we should only be able to see what the character is reasonably able to narrate**. The alternative is heterodiegetic, where the narrator is not a part of the story being narrated. In film or gaming, the narrator would be the omniscient camera moving from place to place to show those parts of the story that are relevant.
- Gaming issues. With 1st person, the choices are always those of the narrator. That can make it difficult to inject the choice of a different character into the game structure. So "Does the bully punch you or kick you?" choice seems odd -- if it's up to me to decide what the bully is doing, why can't I chose to have the bully just run away? In 3rd person, all the choices are being made for "other" people. Even the MC is someone else. In this sense, the choice mechanism is as free as the camera is to move where it needs to go to make the narrative proceed when the game is 3rd person. The choices are heterodiegetic as well.
- Immersion. Obviously 1st person ought to run away with this. What is more immersive than moving a virtual person around the world? Well, that sorta depends. One aspect is the quality of the writing. If "I" am forced to say badly written dialog ("All your base are belong to us"), the immersion is broken. And then there's the problem of the MC being a jerk. Or worse. There's a certain ick-factor intrinsic in the MC being a horrible person (lots of these games require the MC to be what can only be described as a psychopathic serial rapist) and that horrible person is me. Uh, no, I would never do that -- and boom, there goes the immersion.
** There is, of course, a flashback narrative. In this form, the story is being told by a future self, so even though the narrative is 1st person, the narrator has more knowledge then was available at the time the events were happening. That can be an "out" to allow more flexible storytelling. But there are caveats. First, you have to do a flashback -- and for a VN that can have multiple endings, that means the "future" from which the narrator is speaking isn't really defined until the end of the "present" game. And second, even in the future, the narrator isn't going to have perfect knowledge. For instance, "As it turns out, my wife was having an affair with my boss..." leading to a scene in which the MC isn't present, but that "narrator-MC" knows happened, so it's OK to show it. Fine. Did the character learn this during the course of the story being told? If so, there needs to be a scene for it in the game. That locks the storytelling down -- it *has* to happen now. If the MC finds out after the end of the story, great, but how? If it's not shown, the player is going to wonder how it happened, and what happened because of it. Did the MC find out from his wife, from the boss, from someone else who had knowledge? If your story has a choice where then wife and the boss are eaten by aliens, you have a problem. A loose end if there is no effort to explain how the "narrator-MC" learned about it.