Many people assume that the tradition to give Chanukah Gifts/Cards was adopted from the non-Jewish holiday, Christmas, which usually falls out at the same time of the year.
While it is true that the custom, might have changed, in recent years from Chanukah Gelt (money) to Chanukah Gifts/Cards, the source of giving Chanukah Gelt/Gifts is actually rooted in early Halachic works, Jewish law.
Rabbi Chaim Kanievsky Shlita, sites as a source for this custom, the Magen Avraham that states that Chanukah was a time of year when poor people would collect money.
Since people were used to giving out money at that time of year, it became the custom to give one's friends and family to remind them to give charity and good deeds.
The root of the word Chanukah comes from the word Chinuch, which means to educate.
Therefore Chanukah, is a time of year where Jewish communities would focus on matters concerning the education of their children.
Since Chanukah usually falls out in the middle of the school year, gelt/gifts were given to the children as a reward for their scholastic achievements and to encourage them to excel in the coming term.
Another idea revolves around the story of Chanukah itself.
It was the Greek's desire to defile all of the Jew's material possessions and make anything pure impure.
Thus to celebrate our victory we do the opposite.
We take our material possessions (money/gift) and turn them into something holy by giving them in honor of a holiday.
In Keeping with this tradition, here is a great idea for Chanukah Cards which are in fact a Chanukah gift.