I heard a nice Dvar Torah from Rabbi Josh Blass . He posed a question (not the most famous Chanukkah related D.T. question but close - as he pointed out). He called it "the Pnei Yehoshua's question" (which he said is not as famous as the question of why we light for 8 days if the miracle was that the oil lasted for an extra seven days, which he called "the Beit Yoseif's question): Technically they were allowed to use impure oil in this situation (due to the legal rule of Tumah Hutrah Betzibur) so why did they go out of their way to find pure oil?
His original suggestion was that when you're rededicating something, starting fresh, it's of utmost importance to try to regain purity. He shared his memory of being in Ohr Yerushalayim in 1990 and learning in the middle of the night on the top of the water tower. He'd sit there and/or in the empty Beit Medrash and sing songs of dveykut. It was a time so pure.
Chanukah is about reaching back for purity and rekindling our lives with the purest fire we can find.