Although the eight days of Hanukkah are a festive and joyous time, with no fast days and no eulogies if somebody passes away, the laws of mourning do continue through the days of Hanukkah. If somebody passed away before Hanukkah or on Hanukkah, they still observe the shiva (seven-day mourning period). Therefore, people still get together in the house of the mourner to pray together with the avel (mourner). So the question becomes: What about reciting Hallel during the eight days of Hanukkah in the home of a mourner? On the one hand, Hallel is a festive and joyous prayer, which doesn't really fit the sentiment of the grieving going on inside the mourner's house very well. On the other hand, the hachamim (sages) say that a mourner is obligated in all of the mitzvot (obligations) and has to go about fulfilling all of their obligations, including lighting the candles on Hanukkah. Hacham Ovadia concludes as per Maran (Our Master) that Hallel is still recited in the house of the mourner, with the Beracha (blessing) before and after, and the mourner participates.
Only on Rosh Chodesh (new month), where Hallel is a minhag (custom), does the mourner not participate in the Hallel. Sometimes Hallel isn't said at all in the house of the mourner, or the mourners step out while the rest of the kahal (congregation) who are there say the half Hallel. But that's for Rosh Chodesh, which is a minhag. On the days of Hanukkah, when it is an obligation, Hallel is still recited and the mourner participates.
Additionally, if a person is within their 30 days of mourning for a relative or within their year of mourning for a parent, they are permitted to be the chazan and lead the prayers on Hanukkah with the Hallel being recited. Even though they're in a somewhat quasi status during the course of that time, they can lead the prayers for everybody else, especially if the person is the chazan kavua, who typically goes up for the rest of the kahal and wants to be the chazan in order to honor the deceased. So they are allowed to do so on Hanukkah and to lead the tefillah (prayers) for the Hallel.