The short answer is that there are absolutely no rules about giving gifts on Hanukkah. In some families, there is a tradition of giving small gifts on all eight nights. In other families, one large gift is given, or none at all. In some families, children receive gelt (money or chocolate) and not gifts.
Hanukkah celebrates the triumph of the Maccabees over the oppressive Greek King Antiochus who tried to force the Jews to abandon their laws and customs and assimilate into Hellenisitc culture. Because it celebrates an event that took place long after the Torah was written, there is no mention of Hanukkah in the Torah — which is one reason that it is technically a minor Jewish holiday. Its religious status notwithstanding, however, in predominantly Christian countries, largely because of its proximity to Christmas, Hanukkah has been lavished generous attention and the customs that surround it have been influenced by Christmas celebrations.
Originally, Jewish gift giving was associated only with the holiday of Purim, because the Book of Esther states that after Mordechai and Esther saved the Jewish people from the genocidal plot of Haman, the Jews celebrated by giving gifts to one another and to the poor. (Esther 9:22) For well over 1,000 years, Purim was a time to give gifts and Hanukkah was not.
Sometime in the late medieval or early modern period, Jewish schoolchildren started giving secular year-end bonuses to their teachers at Hanukkah. This is the origin of the custom of giving gelt (Yiddish for money). Over time, money was given not just by children to teachers, but also to children themselves so they could use it to play dreidel (another Hanukkah custom influenced by Christmas traditions).
It wasn’t until the late 19th century that Jews started giving children gifts at Hanukkah. This was around the same time that marketers began encouraging a tradition of gift giving among Christians at Christmas. Since then, in many households, Hanukkah gifts have been an important part of the festivities. But from the perspective of Jewish law and tradition, there is no expectation of gift giving.