Are children who are below the age of Bar/Bas Mitzvah to light their own Chanukah candles?
*The Halacha below only applies to those of Ashkenazi Jewry. Sephardim are not accustomed to have any additional family member light the candles, other than the father or leader of the home.
Some Poskim rule, that according to the Ashkenazi custom for every household member to lights the candles, once a child reaches the age of Chinuch he is also required to light. Other Poskim however rule, that even according to the Ashkenazi custom in which all male household members light candles, nevertheless, there’s no obligation to educate a child in a matter of Hidur Mitzvah, and therefore he is not obligated to light. [Practically, according to Chabad custom, male children only begin lighting the Chanukah lights some time before their bar-mitzvah. Nevertheless, in many families of Anash, even very young children are educated to light their own candles. In the words of the Yesod Veshoresh Ha’avoda “One is to be very careful to educate his small sons to light their own Chanukah Menorah.”]
If a child is living on his own: A child who has reached the age of Chinuch [and is living on his own] is obligated to light the Chanukah candles. The age of Chinuch is considered the age that the child understands the Mitzvah of lighting Chanukah candles. [This applies even to Sephardim.]
Daughters living at home:
Daughters who live in the household in which their father is lighting candles, according to the Ashkenazi custom explained in C, are allowed to light candles with a blessing, in addition to the father of the home. Practically, however, the daughters are not to light Chanukah candles in addition to their father, and are rather to fulfill their obligation with their fathers lighting. [Nonetheless, if a girl insists on lighting, and doing so will add to their education in a positive way, then they may do so.]
From the letter of the law, children who are below the age of Chinuch, are not required to be educated to light candles. Regarding Ashkenazi children who are above the age of Chinuch, it is disputed as to whether one is required to educate them to light candles. Practically, the custom today is to educate male children of even very young age to light their own Menorah.
If a child is becoming Bar Mitzvah on Chanukah, when is he to light his candles, before or after nightfall?
Should children who are being educated to light their own candles light with wax candles or with olive oil?
Certainly, from the letter of the law they may light with wax candles, however it is encouraged for one to educate his child to perform the Mitzvah in the most scrupulous manner, and thus preferably olive oil should be used.
 Rama 675:3 “According to our custom that every family member lights their own candles, also a child who has reached the age of education is required to light”; P”M 671 M”Z 1; See M”B 677:13; Kaf Hachaim 677:22; According to Michaber, obviously the child does not light being that according to him only one person lights in each home.
 For the exact age of Chinuch in this regard: See Kaf Hachaim 677:26; 17:10 [Mentions ages: 3, 6, 7, 9]; Admur 343:3 “When he understands the Mitzvah.” [i.e. understands the miracle of Chanukah for which we commemorate the lighting]
 M”A 677:8 “Possibly even according to our custom that every family member lights their own candles, a child is exempt from lighting”; M”B 677:13; Biur Halacha 675:3 “Uledidon”; Shiltei Giborim, brought in M”A ibid; Meiri; Kaf Hachaim 677:22; The M”A and Biur Halacha ibid bring opinions that argue on Rama and hold that even according to the Ashkenazi custom that all male household members light, nevertheless, there’s no obligation to educate a child in a matter of Hidur Mitzvah, and therefore he is not obligated to light. The M”B 675:14 concludes he is only required to light one candle.
 Sefer Haminhagim p. 157 [English] based on Sichas Chanukah 5706 that so is the custom by Beis Harav; This seemingly follows the latter opinion above.
 Shevach Hamoadim p.101; Hiskashrus; Perhaps this also follows a similar change in the later generation for girls to begin lighting Shabbat candles at a very young age
 Shaar 12
 Michaber 677:2; Orchos Chaim
 M”B 677:13; Kaf Hachaim 677:22
 See Olas Shmuel 105; Pischeiy Teshuvah; M”B 675:9; Kaf Hachaim 675:21; Sefer Haminhagim p. 157 [English]; Minhag Yisrael Torah 671:1
 See Poskim ibid; Machatzis Hashekel 677:8 in name of Shiltei Giborim [brought in Sefer Haminhagim p. 157 footnote 623] “The Mehadrin have every family member light candles, whether men or women”; Rambam Chanukah 4 “Whether for men or for women”; Kaf Hachaim 671:16
 Olas Shmuel 105; Pischeiy Teshuvah; M”B 675:9; Kaf Hachaim 675:21; Sefer Haminhagim p. 157 [English]; Minhag Beis Harav, brought in Toras Menachem 5750 2:51; Likkutei Sichos 30:312 “Practically, we do not find that women would light candles on their own”
The reason: Although the Ashkenazi custom is for all household members to light, nevertheless, the custom is for daughters to follow the letter of the law not to light their own Menorah and fulfill their obligation through their father. The reason for this is because women are considered nullified to men in this regard and are hence not included in the Mitzvah of Mehadrin. [Olas Shmuel 105 brought in M”B 675:9] Alternatively, the reason is because women are not expert in the blessings. [Mishmeres Shalom 48:2, brought in Sefer Haminhagim ibid] Alternatively, because they are not expert in the laws of how it is to be done. [Likkutei Sichos 30:312] Alternatively, the reason is because it is improper for daughters to light in face of the mother who is not lighting. [Sefer Haminhagim ibid]
The Chabad custom: As stated above from Sefer Haminhagim, the Chabad custom is for girls not to light candles. Nonetheless, in the year 1988 the Rebbe stated that even girls are to light candles if it will add to their education in a positive way. [Hisvadyus 5748 Vol. 2:91] However, the next year the Rebbe was asked by Neshei Ubnos Chabad if this instruction applies for the coming year as well, and the Rebbe answered that this question belongs to a Rav. The widespread custom today amongst daughters of Anash is not to light candles, as is the custom mentioned in Sefer Haminhagim.
 See previous footnote under Chabad custom!
 Mikraeiy Kodesh Chanukah 11; See Piskeiy Teshuvos 675:4
 Teshuvos Vehanhagos 2:337
 See Likkutei Sichos 12:251