The safest deodorant for a child is a natural one without chemicals. Antiperspirants may also be an option as they can stop the sweat, but it is worth remembering that their bodies are meant to sweat. Antiperspirants may be inappropriate for use in children.
There is some debate as to whether a child should wear deodorant. Try some alternatives first: regular baths and showers, as well as changing bed sheets and clothes regularly. Avoiding spicy and aromatic foods are also things to try. It could be the child is maturing, and hormone changes dictate that a deodorant is the best option.