Is it permitted to allow one's child, who has yet to reach the age of Mitzvah obligation but is old enough to be trained in Mitzvah performance (generally around age 5 or 6, depending on the child's development) to light the Chanukah candles?
The Ben Ish Chai (Rabbi Yosef Chaim of Baghdad, 1835-1909; listen to audio for precise citation) encourages one to allow his child to light the additional candle placed alongside the Chanukah candles – which we generally call the "Shamash" – in order to train him in the performance of Mitzvot. The clear implication, of course, is that one may not allow a child to light any of the actual Chanukah candles.
Others, however, disagree. Rabbi Chayim Palachi (rabbi of Izmir, Turkey, 19th century), in his work "Mo'ed Le'kol Chai," writes that one may allow a minor that has reached the age of training in Mitzvot to light the "Nerot Hiddur," meaning, the candles lit in addition to the single candle strictly required each night. These candles are added for the purpose of "Hiddur," beautifying the Mitzvah, and are not included in the essential obligation, which requires lighting just one candle each night. Therefore, Rabbi Chayim Palachi rules that one may allow a child that has reached the age of Mitzvah training to light these candles. This is also the position taken by Chacham Ovadia Yoseph, in Halichot Olam.
It should be noted that women's obligation with regard to Chanukah candles is the same as men's, and therefore one may allow his wife to light any of the Chanukah candles, including the first candle lit in fulfillment of the essential obligation.
Summary: One may not allow a child under the age of Mitzvah obligation to light the first Chanukah candle, through which one fulfills the essential obligation. He may, however, after reciting the Berachot and lighting the first candle, allow a child that has reached the age of Mitzvah training to light the additional candles, which are lit for the purpose of "Hiddur" – beautifying the Mitzvah.
