Chanukah Gelt

HaRav Chaim Kanievsky, Shlita, explains that the custom to give Chanukah Gelt to children comes from this concept of Tzedaka on Chanukah--putting oneself into a frame of mind to help all those who cannot subsist on their own.
It is said in the name of the Belzer Rebbe that we really must give Tzedakah to the poor in order for them to have money to light the Menorah, to further Pirsumei Nissa.  In order not to embarrass the poor, we freely give to all.

Perhaps the concept of Chanukah Gelt is an outgrowth of the custom of playing dreidel which is often played with money.

The Ponovezer Rav, Rav Yosef Shlomo Kahaneman, Z’tl, said that after the Greeks forbade the Jewish children from learning Torah, their parents had to ‘bribe’ them with gifts to get them to learn once again. Giving gifts on Chanukah commemorates that aspect of the Chanukah miracle. 
Indeed, children should be told when receiving gifts that it is for this reason and that they are expected to increase their Torah learning because of the gifts (see Sifsei Chaim from Rav Chaim Friedlander, Z’tl; Moadim Vol. 2 page 134).

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