From the point of view of a parent, and grandparent, I would say that it would be unwise to let a child of elementary-school age have a computer with internet access in his or her bedroom. We hear a lot of horror stories about youngsters making friends with predatory adults over the internet, and there are undoubtedly infinitely more such stories we never hear about. The simplest way for parents to enjoy peace of mind--at least about this one subject--is not to allow children to have private access to the internet. It might be pretty hard to keep them from having computers, but even those can be harmful if they use them as shortcuts for homework. How can we supervise kids' use of computers when they know more about computers than we do? It seems that everything has a good side and a bad side, doesn't it? Emerson discusses this in his famous essay "Compensation." Schopenhauer spoke about the "law of compensation" even before Emerson. Automobiles are great--but they kill around 30,000 Americans every year. Computers are barely in their infancy, but they are going to turn our whole world upside down.
Speaking from a health conscious perspective, statistically speaking, children with electronics of any kindin their bedrooms are more likely to be obese. Most studies focused directly on televisions, but researchers and pediatricians are now concluding that the pattern is holding true for computers, gaming devices, and cell phones.
So I'm going to say no. I actually don't think electronics belong in the bedroom at all. This is for children and adults. Pick up any magazine in line at the grocery store with an article about stress and sleep, and I can almost guarantee one piece of advice within will say to reserve the bedroom for nothing but sleep. This means no TV, no computer, no phone, no eating.
I'm not as much of an absolutist on this issue as the previous poster is. However, I do not think that there is much of a reason to have a computer in the room of a kid of that age. I very much prefer to have the computer out where the rest of the family can be around so that my kids don't just go off in their rooms by themselves. I like to have more family interaction while it's still possible (before they're teens).
So I would say "no," but more for reasons of familial togetherness and less because of "the evils of the world."
Electronics in the bedroom are kind of a bad idea, effort a children as well as adults. It is correlated with obesity as well as with lack of certain social skills. Personally, I believe that quality of life is enhanced by keeping the bedroom as a kind of safe haven in which the outside world is essentially shut out, and one can be alone with one's own thoughts. Young people especially need such quite time.
I believe elementary school-age children are LESS likely to be tempted to search forbidden web sites than middle or high school-age kids. I don't see a real problem with having a computer in a child's bedroom, although I would absolutely restrict usage to normal hours. I believe children that age should be spending time with the family instead of going it alone on the 'Net in their own room.
There is software that is supposed to block inappropriate content and web sites, but that is not the only reason kids should not have a computer with internet access in the bedroom. The main reason is that it is too tempting. Kids should be supervised any time they are on the internet.