There are a couple of different sub-topics I have found to be the focus of research on video game and computer use by children.
The topic which has received the most attention is violence, particularly the perceived correlation between violent media and violent behavior in those who view violent programming/play violent video games.
As Andy W mentioned, there are a lot of studies on this topic, and again as Andy W mentions, there are claims that the research linking video games and incidents of extreme violent behavior are faulty and fail to acknowledge the significant methodological and constructional divides between existing video game research and acts of serious aggression and violence.
Some studies go so far as to claim that not only is the hypothesis that violent game playing was associated with violent behavior unsupported by the evidence, but focus on the improvements in visuospatial cognition.
Edit: A much more recent study also questions the validity of these studies, while reporting some significant potential benefits:
It turns out that a successful gamer is strategic and technically knowledgeable, and has good timing. Inconsiderate gamers, as well as those who act aggressively or emotionally, generally do not do well. 'The suggested link between games and aggression is based on the notion of transfer, which means that knowledge gained in a certain situation can be used in an entirely different context. The whole idea of transfer has been central in education research for a very long time. The question of how a learning situation should be designed in order for learners to be able to use the learned material in real life is very difficult, and has no clear answers,' says Ivarsson.
'In a nutshell, we're questioning the whole gaming and violence debate, since it's not based on a real problem but rather on some hypothetical reasoning,' he says.
Aggression and violence are not the only negative traits associated with extended computer use/video game play, however.
Computer use in general, and video games in particular, are associated with sedentary lifestyles. In a study of Swiss children, each additional hour of video games played daily was associated with a doubling of obesity risk.. However, that same document also cites a study of Australian children which found that after adjusting for covariates, there was no relationship between obesity and time spent with TV, video games, or computers. The paper then mentions that such null findings are common.
In fact, this same paper lists the results of numerous studies of various age groups and localities, and the results frequently seem contradictory.
Add to that the probability that computer use may not be the primary source of sendentary behavior (according to this somewhat older study, "the impact of computers and video games on sedentary behavior is probably not very large, especially when compared with television, as they together comprise only about 10% of the average daily media budget of children aged two to 18".), and computer use seems unlikely to automatically result in obesity on its own.
Claims of the addictive nature of video games have long been circulating (please note that I include that link only to illustrate the types of claims, as I saw no evidence that that site is remotely reputable, nor anywhere near as "unbiased" as they claim to be, especially since they proudly display "40% of World of Warcraft players are addicted" based off of a single doctor's claim).
I am skeptical of the majority of these claims, as many seem based upon the idea of creating a new definition of video game or internet "addiction" based solely upon the amount of time spent, while others seemingly ignore the possibility of major mental health issues that may have expressed obsessive or violent behavior regarding games or internet usage as a symptom, rather than an underlying cause. However, at least one study has shown that excessive video game playing can result in a number of problems that resemble symptoms of gambling addiction as defined by the DSM-IV.
Attention Deficit Hyperactivity Disorder
This study claims that:
Adolescents who play more than one hour of console or Internet video games may
have more or more intense symptoms of ADHD or inattention than those who do not. Given the
possible negative effects these conditions may have on scholastic performance, the added
consequences of more time spent on video games may also place these individuals at increased risk
for problems in school.
However, this study uses Young's Internet Addiction Scale, which is a tool to measure Internet Addiction Disorder, which is itself not a currently valid disorder, and has not been recommended by the American Medical Association for inclusion in the upcoming DSM-V.