According to the Shulchon Aruch, there should only be one menorah lit at each home. Accordingly, a father has no chinuch obligation to have his child light the menorah. If, however, the child lives in his own home, the father should make sure that he lights the menorah. According to the Ashkenazic custom to have each person light his own menorah, a child should light the menorah as well. There is a minority opinion which maintains that since it is only a stringency for each household member to light, there is no mitzvah of chinuch to train a child to adopt this stringency.
[שו"ע תרעז, ב, ומשנ"ב יג; וראה ביה"ל תרעה, ג, ד"ה ולדידן]