What am I talking about? After all the Beis Yosef was bothered why there are 8 days so why do I want to add another day? The answer is Yom Tov Sheni shel Galuyos. Just like in Chutz Laaretz we keep 8 days of Pesach 2 days of Shavuos, etc. we should add a 9th day of Chanukka.
The Minchas Chinuch in mitzva 301 (where he discusses all the issues of Yom Tov Sheni) discusses this question as well and says that in the time of kiddush hachodesh they actually did keep 9 days of Chanukka because of sefeka d'yoma. He doesn't explain how many candles they lit each night. The Devar Avraham has a similar comment by Sefiras Haomer that in the time of Kiddush Hachodesh they counted twice because of sefeka d'yoma. RYBS did not agree with this at all, he said that you can't count twice that isn't called counting if you have a doubt what day it is. It seems to me that Chanukka is similar according to our minhag where we light the number of candles based on the day. In any case, the Minchas Chinuch states that nowadays we don't because since we really know the date and it is only minhag avosechem, we are mekeil by a d'rabbanan.
The Minchas Chinuch quotes a Ran who seems to disagree. The Ran writes by Megilla that a city where there is a doubt if it was walled from the time of Yehoshua reads on the 14th. The Ran explains as follows. Each day (the 14th and the 15th) is a safek. So in theory we should not have to read at all, on each day say safek d'rabbanan lekula. He says that we can't do this and not be mekayem the mitzva and therefore we read on the 14th and on the 15th we say safek d'rabbanan lekula. The Mishna Lemelech asks from the Ran in Pesachim. The Ran in Pesachim comments as follows. The gemara states that 2 of the 4 cups need haseba and we since don't know which 2 we do haseba for all 4. The Ran asks why not just do haseba on the first 2 and say safek d'rabbanan lekula? He answers why pick the first 2 over the last 2. This seems to contradict his position in Megilla. The Mishna Lamelech answers that there is a difference between not knowing the facts and not knowing what the takana of Chazal was. In Pesachim, we don't know what the takana was. If we pick the first 2 and chazal were really mesaken the last 2 (or vice versa), we are being mevatel completely the takana of Chazal and therefore we need to do all 4. By Megilla we have a factual safek and therefore can say safek d'rabbanan lekula.
Based on this, by Chanukka as well, the Ran would hold that you light on the first day and on the 9th day you would say safek d'rabbanan lekula because again our doubt is in the facts what day is it.
R' Zevin in Moadom V'Halacha brings down a cute answer as follows. The Beis Yosef asked why is Chanukka 8 days after all the miracle was only 7 days? They answer that really Chanukka is only 7 days and the 8th day is because of sefeka d'yoma. The difficulties with this answer are readily apparent but it still is cute.