What will really work to get rid of my child's cough?

Depends. There are different causes of cough and depending on the cause the treatment is different. Examples include asthma, allergic rhinitis, infection. Cough medications are not recommended in children.
Fluids. OTC cough and cold medications really are not effective in relieving cough. Keeping the secretions loose will help the child expectorate them. Honey (with tea) has been found to be as effective as dextromethorphan (a cough suppressant in cough medication, associated with potentially serious side-effects), but should never be given to children under one year of age.