Chanukah is a unique holiday in that the Gemara delineates two extra levels of hiddur mitzvah (enhancing the mitzva) when lighting the candles. The basic mitzva is for the head of the household to light one candle each night for the whole family. However, there is a level of mehadrin where each family member lights a light every night. In mehadrin min hamehadrin each family member lights the corresponding number of candles for that night.
The Beit Yosef discusses a question whether a person who made a blessing on the wrong number of candles must make another blessing when he remembers to light the additional candle(s). He answers that if there was a significant break (approx. 1-2 hours) after the first lighting, one would make another blessing. This is surprising, because in normative Jewish law one doesn’t repeat a blessing on a hiddur mitzva. From this we learn that the mehadrin factor inherent in neirot Chanukah is unique in that it is related to maaseh hamitzva (performance of the mitzva). While there is great importance attached to beautifying a mitzva, such as making a blessing on a fine etrog or tallit, it is only related to mitzva objects with which the person fulfills the fundamental mitzva regardless if the item is beautiful. Therefore, no further blessing is recited. However, when one adds more Chanukah candles, the performance of the mitzva is radically enhanced, it’s intrinsic to the mitzva. It’s not just lighting the candles, but also pirsumei nisa – publicizing the miracle. Therefore, another blessing is recited.
Similarly, the poskim explain that although the basic mitzva of ner ish u’baito, (the father lighting for the household) has already been fulfilled, other family members can still make their own blessing because they are adding to the fundamental mitzva, which is pirsumei nisa.
Can a child who has reached the age of chinuch and is obligated in Rabbinic mitzvot, be motzi (intend to include) an adult with a mitzva d’rabanan such as megilah or neirot Chanukah? The Shulchan Aruch rules that a child cannot be motzi megilah but he could be motzi neirot Chanukah. Rav Soloveitchik explains that megilah is a chiyuv gavra – an adult obligation. Neirot Chanukah is a chiyuv bayit – an obligation on the household. It’s not a transfer from one person to the next. Since a child has an obligation he can automatically be motzi the household.
There’s an old custom to sing Haneirot Halalu as the Chanukah lights are lit. This seems like a hefsek (interruption in the performance of the mitzva). The reason it is not is because it is part of publicizing the miracle.