Milk Foods
The Rema writes that it is proper to eat cheese at Chanukah because of the miracle that was wrought through milk, when Yehudit gave milk to the enemy leader to make him sleep, and thus was able to kill him (Rema SA OC 670:2).
Of course this theme of the righteous woman who entices and overcomes the enemy leader by giving him milk well precedes the Chanukah story. We recognize it from the Biblical story of Yael who overcame the Canaanite general Sisera in this fashion (Judges 4:17-22).
In both these cases, the main significance of milk is not that it can make us sleepy. Rather, there is an element of irony because milk when fed by a woman to her infant is the symbol of loving kindness, but in these cases the milk given as an act of seeming kindness is actual the means of the enemy's downfall. In the story of Yael, this aspect of heaping on kindness is specially emphasized in Devorah's song: "Water he asked, milk she gave him!" (Judges 5:25).
We find a similar idea in the Midrash on the verse "Who would have said to Avraham that Sarah would nurse children?" (Bereshit 21:7). Rashi cites the Midrash which reacts to the word "children", which is surprising since Sarah bore only one child. He writes that all of the leaders of the surrounding people came and that their wives brought their infants but not their wet nurses; Sarah then nursed the princelings of the other nations.
The Midrash explains further that those who came with pious intentions were made righteous by the milk. Those who came with impious intentions (shelo leshem shamayim) benefited in a different way, by being given temporal power as kings and princes; but after they refused to accept the Torah, this blessing turned to a curse and became the reason that this power was removed from them (Rashi and Bereshit Rabba on this verse). Here again we see the same theme: the righteous woman gives of her milk to the leaders of the nations, but when their intentions are wicked this act of loving kindness is their downfall.
This symbolism for milk should hardly be surprising, because this duality is the essence of milk. The gemara (Bechorot 6b) tells us that milk is made from congealed blood. Milk is the most permissible of the animal foods; unlike meat it requires no shechita (ritual slaughter); yet it comes from blood, which is the most strictly forbidden animal food! (As we find in Vayikra 7:27, Devarim 12:23).
Indeed, we find in Chasidic writings that this duality is the basis of the Torah prohibition of milk and meat. Both the milk and the meat are "purged" of their negative bloody nature; in one case the blood is congealed into milk, in the other it is expelled through salting. When the two are brought together, each so to speak "reminds" the other of this sanguine element which has been distanced but not quite eliminated (Likutei Halakhot Breslav, laws of Milk and Meat I:1).
This theme is especially appropriate to Chanukah, which celebrates our victory over the Greek army but even more so over the Greek culture of the Greeks and Hellenists. Unlike other pagan cultures, the Greek culture is considered to be susceptible of assimilation to sanctity, when it occupies its rightful place subordinate to Torah, and doesn't attempt to supplant Torah or to subject it (as was the case with the Septuagint, the Greek translation of the Torah which was likened by the Sages to placing a lion in a cage). For instance, the Torah tells us that "G-d will enlarge Yefet and he shall dwell in the tents of Shem" (Bereshit 9:27). Yefet, the father of Yavan or Greece (Bereshit 10:2) is worthy of Divine blessing but only if he is subordinate to Shem, the progenitor of Avraham. And a Midrash on the commandment of the red heifer likens each description of the heifer - red, complete, without blemish, on which no yoke has rested" (Bamidbar 19:2) to a different nation; "without blemish" refers to Greece (Pesikta deRav Kahana on Chukat).
On Chanukah we eat milk foods, in commemoration of the victory which was wrought when milk, the symbol of loving kindness, was turned to a deadly poison for our enemies. This reminds us that the Chanukah victory gave us the upper hand on a tenacious culture which worshiped beauty and knowledge. These values nourish and elevate us when they are subordinate to G-d's service, but are destructive when separated from it.