Why is the eighth, and last, day of Chanukah called "Zos Chanukah" and what is it significance?
The eighth day of Chanukah contains the Torah reading including the sum total (Numbers 7:84) of all the tribes' leaders' dedication offerings. So the phrase used is "zos chanukas hamizbeach", this was the dedication/inauguration, and the eighth day of Chanukah is then known as "zos chanukah" or "this is chanukah." It has the longest Torah reading of all 8 days.
In Hassidic thought it has significance because it's the wrap-up of the holiday, as if to say "this represents how our chanukah went this year"; in some schools of Hassidic thought it's also seen as an extra-last-last chance for final judgment from the High Holidays (something like "the verdict is written on Rosh HaShanah and sealed on Yom Kippur ... but the book isn't closed until Hoshanah Rabba, and isn't put back on the shelf until Zos Chanukah" ... however you interpret that).
The Bnei Yisaschar (end of Yud) quotes a gematria linking zos chanukah to judgement.
"The eighth day of Hanukah is called “Zot Hanukah” after the daily Torah reading. Rejoicing on “Zot Hanukah” is a minhag of the Baal Shem Tov observed by many Chassidim. (Cf. Likkutei MaHaRiCH, Seder Dinei u'Minhagei Hanukah, p. 714) Teachings of Kabbalah and Hasidism explain that Zot Hanukkah is a day to repent similar to Yom Kippur celebration. It is the final day in which to do teshuva, following Yom HaKippurim andHoshana Rabba, the last chance to get a negative decree dissolved…. Zot has gematria 408, which is 3 times 136, 136 standing each for Tzom/fasting, Kol/prayer, and Mamon/charity, the three things to nullify an evil decree. Instead of chest beating, be a Hassid and dance! After all Aharon the Kohen corresponds to the Man of Hesed, lovingkindness, and it is to he and his vestments do we now turn our attention." From Sefer chadash Yamenu - R. Rachmiel Drizin [Shlit’’a]
Rabbi Elimelech Biderman shlita writes (p.12) in his Torah Wellspring (Parshas Mikeitz 5781) as follows:
The Yismach Yisrael (Chanukah 53) writes that when the Torah says, זאת, it refers to something one can see with his eyes. The Yismach Yisrael explains that on זאת חנוכה tzaddikim can see all of the holy lights that shone throughout the days of Chanukah. The kedushah is very great on this day, and tzaddikim can see it. Therefore, it is called זאת חנוכה, something that can be pointed to and seen with one's eyes. "But this revelation is only for the perfect tzaddikim. It states (Tehillim 118:23), מאת ה' היתה זאת, we believe that Hashem gives this revelation on זאת חנוכה, however, היא נפלאת בעינינו, we aren't able to see it… Although we don’t see or feel this special revelation, nevertheless, we are happy for the tzaddikim, for we believe that they see it. We are happy with their joy…"
בזאת ידעתי כי חפצת בי (Tehillim 41:12) The Yismach Yisrael explains בזאת, since you gave us זאת חנוכה, I know that You desire me and that You want to grant me good.