While the only mandated mitzvot for Chanukah consist of lighting candles and saying the full hallel, there are numerous other customs that have come to be associated with Chanukah.
The custom to play driedel on Chanukah is steeped in mystical allusions. From the letters which appear on the driedel to the way the driedel spins, people have offered explanations to link this to Chanukah. The Beni Yisscar, R. Tzvi Elimelch says that the reason the dreidel is spun from the top and the gragger on Purim is turned from the bottom has to do with how each holiday's miracles were effected. On Chanukah the miracle came from above, directly from God. However, on Purim, the miracles were directly brought about by the actions of Ester, Mordechi and the Jewish people. Thus, the dreidel is spun from the top showing the miracle came from above, and the gragger from the bottom showing the miracle came from below. Others explain the symbolism of the letters that appear on the dreidel, נ, ג, ה, ש. According to one explanation these hint to the mitzvot that we have on Chanukah, נרות שמונה (candles all eight nights) and הלל גמר (the complete hallel). Others note the gematria (numerical value) of the letters which correspond to the same gematria as משיח (the Messiah). Others still, link the letters with גשנה the city Yosef secured for his family in Egypt.
According to at least one source, the custom of playing dreidel was actually started in the time of the Maccabis. They say that in an effort to circumvent the Greek decree against studying the Torah, children and their teacher would have a dreidel handy to start playing in case the Greeks came upon them studying the Torah. They would claim they were not studying instead they were just playing dreidel.
Despite all of these explanations, in truth, dreidel is not Jewish in origin. Rather, driedel is really the rather old game of teetotum. Teetotum, which uses a top with four sides and four letters is one and the same with dreidel. The letters that appear on the dreidel are really just the Hebrew letters that appear on a German or Yiddish teetotum, G, H, N, S. G= ganz (all), H halb (half), N nischt (nothing) and S schict (put). Teetotum dates back to at least the 16th century long before we have any Jewish allusions to dreidel(it was originally totum or top, but became TEEtotum due to the use of T for take all, on the top). The well-known depiction of children's games done by Brueghel in 16th century includes Teetotum(see here and here for the complete painting). The earliest Jewish mention of dreidel or the significance of it dates to the late 18th century.
The story connecting dreidel to the ruse of the Maccabis was first published in the book Minhagi Yeshurun, which was first published in 1890 (the name was changed to Otzar Kol Minhagi Yeshurin in the third edition, which is available online here from Hebrewbooks.org . The author included a nice picture of himself at the beginning, although he was a Rabbi in Pittsburgh at the turn of the twentieth century, he is holding a quill pen.) His source is a contemporary of his. [As an aside, although his explanation of dreidel is well-known he offers a similar explanation for playing cards on Chanukah, i.e. that the Maccabi did so. However, that one is not nearly as well know.]
The custom of Chanukah Gelt appears to have changed over time. The earliest sources that mention money on Chanukah connect it with either collecting money for the poor (presumably for money to purchase the necessary implements for the Chanukah lights)(Sefer Mataamim) or giving money to ones children's teachers. (Hemdat Yamim, Chapter 3 Chanukah early 18th century, anonymous author, some claim was Nathan of Gaza, Shabbtai Tzvi's "prophet" others just a student of the Ari).
Again, especially amongst the Hassidic commentators, the custom took on a life of its own, both in its scope to include giving money to children and in its significance. There was also a special emphasis on giving their respective Rebbi money as well. R. Chaim Palache (Pellagi) (1788-1869) is the first to mention giving children money. He offers a kabbalistic reason "as children are representative of נצח והוד (eternity and glory). Something I don't profess to have any idea what that means.
Another custom, again somewhat late in origin, is the custom to not study Torah on Christmas eve. Menachem Butler has a post here on some sources, however, one should add that there is now a full length sefer devoted to this topic, Yisrael Barukh Mestinger, Nitel U'Meorosav, 2000. As well as a pamphlet, Hefaru Toresecha, maamar maktif mminhag avotanu bi-yadun (sic) odot lel ha-ofel nitel nacht, u-minhag yisrael l'vatel ma-asek ha-torah, 2004. Additionally, R. Gavreil Zinner, devotes a section of his work, Neta Gavreil on Chanukah to Nitel.
For more on the various customs associated with Chanukah, see Neta Gavreil Chanukah; Pardes Eliezer Chanukah 2 vol.; R. Yitzhak Tessler, HaDreidel (Sivvivon) B'Chanukah: Mikoroteha, Tameha, u'Minhageha, in Ohr Yisrael 50-62, vol. 14 (Tevat תשנ"ט).
The custom to play driedel on Chanukah is steeped in mystical allusions. From the letters which appear on the driedel to the way the driedel spins, people have offered explanations to link this to Chanukah. The Beni Yisscar, R. Tzvi Elimelch says that the reason the dreidel is spun from the top and the gragger on Purim is turned from the bottom has to do with how each holiday's miracles were effected. On Chanukah the miracle came from above, directly from God. However, on Purim, the miracles were directly brought about by the actions of Ester, Mordechi and the Jewish people. Thus, the dreidel is spun from the top showing the miracle came from above, and the gragger from the bottom showing the miracle came from below. Others explain the symbolism of the letters that appear on the dreidel, נ, ג, ה, ש. According to one explanation these hint to the mitzvot that we have on Chanukah, נרות שמונה (candles all eight nights) and הלל גמר (the complete hallel). Others note the gematria (numerical value) of the letters which correspond to the same gematria as משיח (the Messiah). Others still, link the letters with גשנה the city Yosef secured for his family in Egypt.
According to at least one source, the custom of playing dreidel was actually started in the time of the Maccabis. They say that in an effort to circumvent the Greek decree against studying the Torah, children and their teacher would have a dreidel handy to start playing in case the Greeks came upon them studying the Torah. They would claim they were not studying instead they were just playing dreidel.
Despite all of these explanations, in truth, dreidel is not Jewish in origin. Rather, driedel is really the rather old game of teetotum. Teetotum, which uses a top with four sides and four letters is one and the same with dreidel. The letters that appear on the dreidel are really just the Hebrew letters that appear on a German or Yiddish teetotum, G, H, N, S. G= ganz (all), H halb (half), N nischt (nothing) and S schict (put). Teetotum dates back to at least the 16th century long before we have any Jewish allusions to dreidel(it was originally totum or top, but became TEEtotum due to the use of T for take all, on the top). The well-known depiction of children's games done by Brueghel in 16th century includes Teetotum(see here and here for the complete painting). The earliest Jewish mention of dreidel or the significance of it dates to the late 18th century.
The story connecting dreidel to the ruse of the Maccabis was first published in the book Minhagi Yeshurun, which was first published in 1890 (the name was changed to Otzar Kol Minhagi Yeshurin in the third edition, which is available online here from Hebrewbooks.org . The author included a nice picture of himself at the beginning, although he was a Rabbi in Pittsburgh at the turn of the twentieth century, he is holding a quill pen.) His source is a contemporary of his. [As an aside, although his explanation of dreidel is well-known he offers a similar explanation for playing cards on Chanukah, i.e. that the Maccabi did so. However, that one is not nearly as well know.]
The custom of Chanukah Gelt appears to have changed over time. The earliest sources that mention money on Chanukah connect it with either collecting money for the poor (presumably for money to purchase the necessary implements for the Chanukah lights)(Sefer Mataamim) or giving money to ones children's teachers. (Hemdat Yamim, Chapter 3 Chanukah early 18th century, anonymous author, some claim was Nathan of Gaza, Shabbtai Tzvi's "prophet" others just a student of the Ari).
Again, especially amongst the Hassidic commentators, the custom took on a life of its own, both in its scope to include giving money to children and in its significance. There was also a special emphasis on giving their respective Rebbi money as well. R. Chaim Palache (Pellagi) (1788-1869) is the first to mention giving children money. He offers a kabbalistic reason "as children are representative of נצח והוד (eternity and glory). Something I don't profess to have any idea what that means.
Another custom, again somewhat late in origin, is the custom to not study Torah on Christmas eve. Menachem Butler has a post here on some sources, however, one should add that there is now a full length sefer devoted to this topic, Yisrael Barukh Mestinger, Nitel U'Meorosav, 2000. As well as a pamphlet, Hefaru Toresecha, maamar maktif mminhag avotanu bi-yadun (sic) odot lel ha-ofel nitel nacht, u-minhag yisrael l'vatel ma-asek ha-torah, 2004. Additionally, R. Gavreil Zinner, devotes a section of his work, Neta Gavreil on Chanukah to Nitel.
For more on the various customs associated with Chanukah, see Neta Gavreil Chanukah; Pardes Eliezer Chanukah 2 vol.; R. Yitzhak Tessler, HaDreidel (Sivvivon) B'Chanukah: Mikoroteha, Tameha, u'Minhageha, in Ohr Yisrael 50-62, vol. 14 (Tevat תשנ"ט).