Jewish tradition gives some insights as to why we play a dreidel on Chanukah. The four letters on the dreidel, for instance, spell out "נס גדול היה שם," which means "A great miracle happened there." The dreidel was a facade that Jews in that era would use to cover up the fact that they were studying Torah [since Torah study was illegal during this period]. The gematria, or Jewish numerology, adds the numeric values of the four letters, which comes out to 358. 358 also happens to be the numeric value of the word משיח (Messiah), meaning that when the Messiah comes, we will not have to look at one facet, but will be able to look at all four at the same time.
As insightful as this all might be, the dreidel itself is a paradoxical symbol. One of the important lessons of Chanukah is to not give into cultural assimilation. Yet the tradition teaches us that the Jews of that time played it as to not arouse suspicion of studying Torah. One has to keep in mind that Jewish practices as a whole were banned in this society. If playing the dreidel game were a particularistically Jewish thing to do, they would have been arrested on sight. The fact that they used the dreidel in a Greek-ruled society to cover up Torah study means that playing with a four-sided top was something that the Greeks did. As Rabbi Golinkin points out, "in order to celebrate the holiday of Hanukkah, which celebrates our victory over cultural assimilation, we play the dreidel game, which is an excellent example of cultural assimilation."
Rather than dismissing this as a hypocrisy in Jewish practice, we can derive an important lesson here. Rather than teach us a conflicting message of dichotomy, we can take the lesson that the dreidel represents the importance of cultural integration. Back in Biblical times, the Israelites were influenced by the sacrificial system, a system which existed prior to the Exodus out of Egypt. It was a universal mode of worship, and the Israelites used it until there was no more Temple. Maimonides was the greatest post-Biblical influence on Jewish thought. Maimonides' most influential teacher was Aristotle, who was a Greek who happened to not be Jewish. The notion of marriage was even influenced by their respective rulers. Under Christian rule, the Ashkenazi Jews adopted marriage between "one man and one woman," whereas under Islamic rule, the Sephardic Jews permitted polygamy because that is what the surrounding culture did. The fact that there is a difference between Sephardic and Ashkenazic cuisine implies the influences of the surrounding cultures on Jewish gastronomy. The melody for the most popular Chanukah song, Maoz Tzur, is based off a 14th-century German folksong. Klezmer music was an adaptation of secular, Eastern European music, most notably that of the Romanians. The Yiddish language is another example of how the Jews have taken from other cultures and adapted to it. After all, Yiddish is a fusion of Hebrew, German, Aramaic, and a bit of some Slavic languages.
Although I can come up with more examples, it suffices to say that Judaism does not live in a bubble. One of Judaism's redeeming qualities is its ability to adapt, and that includes on a cultural level. Adaptability, and thus integration, have contributed to Jewish survival. Not only has it kept the Jewish people alive, but the capaibility to take good ideas from other cultures is what has made Judaism stronger.