The tradition of giving money to children during Hanukkah is called ‘d’mey Chanukah’ or ‘maot Chanukah’ (Chanukah gelt). An ancient tradition tells us that this custom dates back to the days of the Maccabees (see below) who liberated Jerusalem from the hand of the Greek. During their domain over the land the Greek imposed their culture over the Jews to the extent that there was a need to re-educate the children about the Torah and Jewish life. There was a need for many teachers and obviously there was the need to pay the teachers. The pupils were those who brought the money gifts to the reachers.
This is the strong link between Chanukah and Chi•nuch, education. (you read the ‘chi’ as hi•nuch and not as the ‘chi’ as in the word ‘chill’, for example). Both words share the same, identical root. Perhaps this story best demonstrates the connection.
More common is the belief that the custom had its origin in the 17th-century practice of Polish Jewry to give money to their small children for distribution to their teachers. In time, as children demanded their due, money was also given to children to keep for themselves. Teenage boys soon came in for their share. According to Magen Avraham (18th century), it was the custom for poor yeshiva students to visit homes of Jewish benefactors who dispensed Chanukah money (Orach Chaim 670). The rabbis approved of the custom of giving money on Chanukah because it helped to spread the story of the miracle of the oil. According to popular legend, it is linked to the miraculous victory of the Maccabees over the ancient Greeks. To celebrate their freedom, the Hasmoneans minted national coins, [The most famous was called the ‘Sela’ coin]. It may also have begun in 18th-century Eastern Europe as a token of gratitude toward religious teachers, similar to the custom of tipping service people on Christmas. In 1958, the Bank of Israel issued commemorative coins for use as ‘d’mey Chanukah,’ ‘Hanukka gelt’. That year, the coin bore the image of the same menorah that appeared on Maccabean coins 2,000 years earlier.