What significance does the name "Chanukah" have to the holiday? There are many different approaches to this question.
An answer mentioned by the Kol Bo, Avudraham, Tur, and Ran is that the Chashmonaim rested (Chanu) from fighting with the Greeks on the 25th (in Gematria, Kah) of Kislev. Chanukah thus is a compound word with both of these ideas. The Noam Elimelech believes that Chanukah actually is based on the Hebrew word Chein, grace; Hashem showed the Jews Divine Grace on the 25th of Kislev.
The Or Zaruah and Maharshah (Shabbat 21b) observe that the name Chanukah sets up the parallel to the Chanukat HaMizbeiach, which is particularly relevant to the holiday of Chanukah. As the Gemara states in Masechet Avodah Zarah (52b), the Chashmonaim stored the old Mizbeiach (because it was now invalid due to its being made Tamei by the Syrian-Greeks) and built a new one. Also, on every day of Chanukah we read from the section about the Chanukat HaMizbeiach in Parashat Naso. Rav Yaakov Emden suggests that in addition, Chanukah represents the Chanukat HaBayit, which happened on approximately the same date many years earlier.
The Avudraham, Ateret Zekeinim, and Pri Megadim all point out that Chanukah is an acronym for the Halacha "Chet Neirot VeHalacha KeVeit Hillel," "Eight candles, and the Halacha follows Beit Hillel (who opined that we start with one candle on the first night and add one every night; Shabbat 21b)."
These are a few of the reasons why the Chag that we celebrate at this time of year is known as Chanukah.
-Adapted from The Artscroll Mesorah Series on Chanukah