One thing you will learn with a videogame is discipline. Surprised? No reason. Imagine you're 5 and you're playing this game. Suddenly, you come face to face with an obstacle. It's a hole, and you fall in it, you lose a life. You fail to jump over it a few times, and then it's Game Over. No matter how angry you will get, the game will remain as intransigent as a block of concrete. Even if you cry and threaten to destroy it, the game will remain pitiless. Unlike a parent, a game will never compromise: you either play by the rules or you stop playing. Children are quite spoilt these days and so the confrontation with videogames is a good experience in that area. The child is alone with the game, and has to fend for himself because there is no other way to get through things than to persevere and work.
Yes, I said work. If you know anything about videogames, you know that many of them are difficult and time-consuming. That was especially true of older games; nowadays, they are longer, but simpler (or so it seems to me, as a trend). I mentioned kids but it's the same with adults. I know many adults who simply don't have the patience to sit through a videogame and finish it, or gather the perseverance required to beat the game. I've seen things being destroyed because the game was too hard and the player too impatient. Videogames do reveal things about ourselves.
So, discipline is one important thing with videogames, because videogames are pitiless. Videogames don't give a rat's ass whether you're crying your eyes out and spending your sleep-time on it. Videogames don't care about you more than most of the world does, and the sooner you realise this, the better for you. And with this comes perseverance: you cannot bribe a game, you can only win through the rules established by the game itself.
The other thing that makes videogames a pedagogical tool of quality is this: when playing, the player learns to connect elements. Depending on the game, you will have to do a lot of connecting, and thus you will train your brain in matching elements, much in the way that you are asked to do on IQ tests. Of course, the level at which you will be asked to connect things will vary greatly whether you're playing Tetris or Myst or Doom, but in either case, you will make your brain work a lot about connections. Any game makes you work on connections: even if you play Super Mario Bros. you'll still have to coordinate the jumps and everything. This is a lot more work than it seems, and if you intend to go far in the game, you will have to muster serious skills.
So reflexes are improved by videogames, as well as character (depending on your natural character; I never saw any impatient person grow more patient). Just like any task, you learn your limits and strengths and weaknesses by testing yourself. Whether it's an essay you must write or a videogame you want to beat, it remains a task that is difficult to do. Personally, I have played more games that were unnervingly difficult than I worked on essays.
Now this is not going to apply to today's games. As a kid, all the games were in English, and in order to understand a little bit of what the game was saying, we had to do some translating. My family didn't know a word of English. I was 5, so I didn't know any either. We bought a dictionary for English/Français and tried to make do with that. I eventually learned some words of English that way at the age of 5, which may not be all that extraordinary for evolved countries, but in my country, you don't get to learn English before the advanced age of 12 or 13.
Another thing I am forever grateful to videogames for is the music. If you have ever played games like Zelda or Megaman, you know what I'm talking about. Not to mention Castlevania's soundtrack, all of which are covered by countless musicians all over the Internet. Those songs were killers, and for a child to be exposed to such masterful music is a truly awesome experience, which I remember with much nostalgia. I still love those songs today, and I still believe they're amazing; some of them are downright genius. Who doesn't know the Super Mario Bros. tune? That stuff is as famous as the Star Wars theme. So thanks to videogames for my introduction to music, because, when I was 5, there was no music in my family.
Any art is good because it works on expending your mind. What does that mean? It just means that whenever you are confronted with something new, you get to work out new ways to deal with it. Whether it's a novel, a videogame, an essay, a painting, a movie, etc, doesn't matter. You're still stimulating your mind and brain through any of these.
So these were the good things in videogames that you don't hear about all that often, but that every gamer knows about, consciously or not. Of course, the more obvious aspects of videogames I didn't need to mention, such as having fun. Games aren't like work in that you're not paid for playing them, but they're not like mere entertainment in that they are very demanding, and you will rarely have only 100% fun playing them. It'll still be fun, but the way good work is; that is, you're proud of having accomplished something. It's no fun replaying the same level 40 times in a row and still losing, but the point isn't merely fun, that's why I say it's not just entertainment like a TV show can be. There's work to do, and it only depends on you.