I just finished reading a recent review by Dapne Bavelier about the benefits of playing action video games. This review does not address the controversial issue of whether video games cause people to become more violent or not (I am personally on the fence on that one).
Rather, this review focus on the effects on more basic perceptual processes, and how playing video games affects these abilities. All this is put within a "learning to learn" framework. That is, will playing action video games help you find patterns or gain knowledge when you are not playing video games as well.
The pattern is very clear here. Video games will help you in a number of different ways. People who play action video games (sim city or civilization won't do), are faster at identifying objects, without making more errors, there are not as easily distracted by irrelevant stimuli, and they have greater contrast sensitivity. It seems as though video games enhances eyesight in general.
There are also substantial benefits in certain cognitive functions. People who play action video games enhance their ability to perform mental rotations (ex determining which 3D shape matches another 3D shape). Mental rotation in turn correlates with improved mathematical skills. Multi-tasking is another cognitive function that is improved in video game players.
Significant improvements have also been seen in decision making and reaction times. These factors may determine whether you brake in time to prevent a traffic accident or whether you die.
Thus in general, playing action video games improve a broad range of perceptual/cognitive abilities which probably helps the individual find relevant information in the environment, even though that environment does not involve war.
The objection which I am sure that some of you have thought of already is the classic "correlation does not equal causation". Maybe people who are good at all of these perceptual/cognitive tasks to begin with like video games more and therefore play more. Indeed, it you are slow, you get killed a lot in many of these games which can be very discouraging (I have played a little bit online and it sucks when you get shot 30 times over by a ten year old).
Researchers of course anticipate this objection and therefore conduct controlled studies. For instance you can take a random group of student with equivalent video game experience and then pay them to play either call of duty or civilization (a turn based game) for a few hours per day for a week or so. Differences between these groups after the intervention are probably due to the different treatments. Such controlled studies support the conclusions above, video games really do improve perceptual/cognitive functions.
So depending on whether you think action video games will increase aggression (like I said, I think the jury is still out on that one), if your child is playing violent video games and still not missing out of school etc, then you might even want to encourage this.
Bavelier D, Green CS, Pouget A, & Schrater P (2012). Brain plasticity through the life span: learning to learn and action video games. Annual review of neuroscience, 35, 391-416 PMID: 22715883