The Chesed L'Avraham (Rebbe Avraham of Radomsk zt'l) states that Zos Chanukah is mesugal for hastening the redemption because this day we read וזאת חנוכת המזבח ביום המשח אותו (Bamidbar 7:84) and המשח is similar to the word משיח.
Another hint to משיח is from מדליקין שמונת ימי חנוכה. The first letters spell משיח.
The gematriya of the letters גנש"ה on the dreidel is משיח.
It states in this week's parashah (41:14) וישלח פרעה ויקרא את יוסף ויריצהו מן הבור "Pharaoh summoned, and he called Yosef, and he was rushed out of the dungeon." The Seforno writes, "[Yosef was saved quickly, as it states ויריצוהו, they rushed him out], like all of Hashem's salvations that happen quickly and immediately. As it states (Yeshayah 56:1), כי קרובה ישועתי לבוא, 'My salvation is near.' This is what happened in Mitzrayim, as it states (Shemos 12:39) כי גורשו ממצרים, 'they were banished from Mitzrayim.' Chazal tell us that their dough didn't have time to rise before the King of kings, Hakadosh Baruch, redeemed them. This will also occur in the future, as it states (Malachi 3:1) ופתאום יבוא אל היכלו האדון אשר אתם מבקשים, 'Suddenly, he will come to his heichal, the master (Moshiach) whom you await.' May it be speedily in our days.