If a child is diagnosed with meningococcal disease, can anything be done to protect the other children with whom he has contact?

Antibiotics. Prophylactic antibiotics can be used for close contacts to prevent secondary infections.
Yes. Sometimes antibiotic prophylaxis and/or vaccination is recommended to others in contact with a child who has Meningococcal meningitis. Talk with your physician and s/he can check with the local health department if necessary.