Should children eat mediumwell or rare meat?

Medium-well, certainly. Thoroughly cooking meat reduces the risk of contracting a food-borne pathogen, such as Salmonella or E. Coli, which can cause vomiting and horrendous bouts of diarrhea. Young children are especially vulnerable to food poisoning, and are among the most likely to die of such an illness. Admittedly, the probability of contracting a food-borne illness from undercooked meats is relatively small, but there's no reason to be gambling with the lives of children.