There are two minghagim for the Torah reading. The Encyclopedia Talmudic gives a good discussion. However, I see that some sidurim say that in Eretz Yisrael the Yisrael repeats the day while in the Galut, Yisrael reads the the next day. Any idea where this idea got started? Someone confused the reading for Sukkot??
Shalom, Thank you for your interesting question. You are correct that there are two customs about what the third aliyah for Hanukah is - either repeating the reading the Cohen and Levi read of the offering of the Nasi for that day (i.e. rereading the section in Parshat Nasso that talks about the second day on the second day of Hanuka etc), or that the third aliyah continues directly where the Levi left off, and reads the next verses that talk about the next day. In Eretz Israel the custom is, as you said, to reread the section that talks about the day of the offerings that is the same day as the day of Hanukah. This argument is found in the Shulchan Aruch (Orach Haim 684,1). The Shulchan Aruch says to repeat the section of that day, and the Rema says to keep reading onwards with the verses of the next day. The custom in Eretz Yisrael today, even amongst the Ashkenazim, is like the Shulchan Aruch (See Rav Tukachinski's work Sefer Eretz Yisrael, section 13). The basis for this argument is found in the Rishonim, and the Tur is of the rereading opinion, and the Haga'ot Maimoniot is of the opinion to continue to the next day (see the Tur and Darchay Moshe ibid). The Vilna Gaon, in his glosses to the Shulchan Aruch (ibid) writes that the argument is based on how to apply the rule that we only read a section from the Torah on festivals that talks about that day's festival - hence the opinion to reread that day's section. However the other opinion holds that the next day's offerings are not considered as totally irrelevant to Hanukah, and so we continue to read the next verses. He brings sources in the Talmud for these arguments, but his conclusion is that the opinion to reread the day's verses. This probably explains the custom in Eretz Yisrael today, where many Ashkenazi customs follow the Vilna Gaon instead of the Rema, who is followed more closely outside Israel. I hope this information helps you - Chag Same'ach!