There is a custom to eat cheese and dairy products during the holiday of Hanukah to commemorate the heroism of Yehudit, a Jewish woman who, during the struggle against the Greek persecution, was abducted by a Greek official named Eliporno. Yehudit fed the official large amounts of cheese to make him thirsty, at which point she served him wine. As a result of drinking wine the official fell into a deep sleep, which enabled Yehudit to kill him, marking an important step towards the Jews' eventual triumph over the Greek oppressors. In commemoration of this great miracle, the custom developed to partake of cheese and other dairy products throughout Hanukah. One should not treat this custom lightly as it is well-established in authoritative Halachic sources.
An interesting question concerning Berachot arises in the common case of one who places cheese on top of a cracker. The famous principle of "Ikar Ve'tafel" establishes that when two foods are eaten together, one recites only the Beracha required for the primary food, and this Beracha covers the subordinate food, as well. In this case, then, when one partakes of a cracker together with cheese, he would recite the Beracha of "Bore Mine Mezonot" as the cracker constitutes the primary food. This Beracha covers both the cracker and the cheese – which is seen as subordinate to the cracker. Similarly, if one places cheese on a rice cake, he would recite the Beracha of "Bore Peri Ha'adama" over the rice cake, and this Beracha covers both the rice cake and the cheese.
The question arises as to whether one must recite a new Beracha if some cheese remains after he finished eating the cracker. Must he recite the Beracha of "She'hakol" over the cheese as now he eats it independently, and not together with a cracker? Or, perhaps since the entire piece of cheese had been covered by the Beracha of "Mezonot" recited over the cracker, even after one has completed the cracker the cheese does not require a new Beracha.
This issue is subject to a debate among the Rishonim (Medieval Halachic scholars). The Rashba (Rabbi Shlomo Ben Aderet of Barcelona, 1235-1310) was of the opinion that in such a case one must recite a new Beracha, since he now eats the subordinate food independently, rather than together with a primary food. The Ritva (Rabbi Yom Tov Ben Avraham of Seville, 1250-1330), by contrast, held that once the subordinate food was covered by the Beracha recited over the primary food, it no longer requires a Beracha even if it is eaten independently.
Which of these views does Halacha follow?
The Mishna Berura (commentary to the Shulhan Aruch by Rabbi Yisrael Kagan, the "Hafetz Haim," 1839-1933) addresses this question amidst his discussion of a certain kind of jelly that was customarily eaten on a cracker (168:46). This particular jelly was actually regarded as the primary food, and the cracker the subordinate food, and thus one would recite a Beracha over the jelly which covered both the jelly and the cracker. The Mishna Berura raises the question of whether one must recite "Mezonot" over a small piece of cracker than remained after one had eaten all the jelly. He rules (listen to audio for precise citation) that one would not, in fact, recite a Beracha over the remaining piece of cracker, since the cracker was eaten mainly with the jelly and was thus covered by the Beracha recited over the jelly.
It indeed appears that the accepted custom is not to require an additional Beracha over the portion of a subordinate food that remains after the primary food has been completed. Although Rabbi Moshe Halevi (Israel, 1961-2001) actually does require reciting a Beracha in such a case, the authorities with whom I have consulted informed me that the accepted practice is not to recite a Beracha.
Summary: There is a widespread practice to eat cheese and other dairy products throughout the festival of Hanukah, and one should not treat this custom lightly. If one eats cheese on a cracker, he recites the Beracha of "Mezonot" which covers both the cracker and the cheese. Even if some cheese remains after he completes the cracker and he eats this remaining cheese independently, he does not recite a new Beracha over the cheese, as it had been covered by the "Mezonot" he had recited over the cracker.
