While I rarely encounter them myself, there are a large number of people out there who refuse to let their children play videogames (along with roleplaying games, and various other sorts), believing it will have a negative effect on them. I wish I ran across such people more often, as I have had ample opportunity to research this subject, and I would very much like to explain to these people how they are stunting their children's mental growth through these restrictions. While most people reading this essay presumably share my mind set on this subject, I am hoping they will share it with those who might benefit from reading it. Here are just some of the many benefits a childhood (or for that matter an adulthood) spent playing videogames can yield.

1. Hand-Eye Coordination: This one is something of a given I'd hope. Various studies over the years have proven that playing videogames (particularly those from the heyday of the arcade) can dramatically improve one's hand-eye coordination and reflexes. The majority of videogames focus a great deal on this, requiring progressively quicker and more accurate responses.
2. Fairness: When playing other types of games with a small child, people will generally be inclined to let them win. Doing otherwise would presumably mean the child would never win, making you feel guilty, but will leave the child with false confidence in their own skills. A videogame however can't pull punches like this, giving the child a fair assessment of their own skill, and ensuring they are mentally prepared to lose games in the future.
3. Perseverance: With rare exception, one only loses a videogame when one makes a mistake. It's generally obvious after the fact where the mistake was made, allowing one to avoid it next time, and progressively improve over their past attempt.
4. Pattern Recognition: Nearly every videogame features at one point or another, obstacles which move in a complicated pattern. Progress can only be made by recognizing this pattern and devising a pattern of action of one's own to follow based on it.
5. Navigation: A great deal of games take place primarily or entirely within giant mazes. Navigating around in these provides many useful skills, such as identifying landmarks, developing a good sense of direction, and the skill sets needed to explore methodically. More practically, a child who spends a great deal of time exploring mazes in games is far less likely to get lost in an unfamiliar city (or neighborhood, school, patch of woods), and if lost already, they will be far more likely to find their bearings and reach familiar ground.
6. Abstract Logic: A good number of videogames, particularly those of the Adventure genre, feature surprisingly complex logic puzzles, typically requiring a great deal of thought and creativity to solve. This builds skills that are useful in every facet of life.
7. Literacy: A great deal of videogames, particularly those of the Roleplaying (or RPG) genre frequently require players to read large bodies of text to gain clues on what must be done next to progress through the game, and/or to advance the story contained within the game. Aside from the obvious benefits for children, many adults have been known to practice a foreign language with the help of imported games.
8. Mathematics: RPGs and strategy games require a great deal of mathematical calculations, often on the spur of the moment. Dividing a numerical expression of how much damage a character can take by how much they are damaged from each of a monsters attacks to determine how long they can go without being healed. Calculating how best to distribute resources in a city for it to operate at peak efficiency. Things of this nature.
9. Elimination of Violent Tendencies: Many people condemn violence found in movies, television, and games, claiming that it inspires actual violence. This logic has no basis in reality, and in fact, the opposite is true. All people have innate aggressive tendencies, which must be expressed in some form or another. Some express these tenancies by committing actual violence, be it simple rough-housing or something more serious. Others seek cathartic release by watching violence in entertainment. Still others relieve their aggressive tenancies through other forms of activity, such as various games and sports. A videogame provides the active expression of this last option, and in some cases can also include the same sort of cathartic release as watching a violent work of fiction. Personally speaking, I find playing videogames to be quite satisfying in this regard, and in a lifetime spent with easy access to them, I have never been at all inclined to bring harm to anyone.
10. Empathy: Many videogames particularly those which require a great deal of abstract logic, feature an in-game protagonist with a unique personality and general skill set. Determining how such a character would approach various problems and acting accordingly makes it much easier to progress through the game, and at the same time helps the player learn to empathize with those of vastly different backgrounds from their own.