It is forbidden to deliver eulogies during Hanukah. The exception to this rule is a eulogy delivered for a Torah scholar at his funeral, which may be delivered during Hanukah. A eulogy for a Torah scholar may also be delivered on Hanukah when the news of his death first arrives, even if this occurs well after his passing (based on Shulhan Aruch – Yore De’a 401:4).
It is similarly forbidden to observe a fast during the holiday of Hanukah. Even in communities that follow the custom to fast on the anniversary of a parent’s death, one whose parent died during Hanukah may not fast on that day.
In a case where one’s family member passed away on Hanukah, Heaven forbid, before the burial he has the status of "Onen" and therefore should not light the Hanukah candles. His wife or other member of his household should light the candles in his stead, with a Beracha. If, however, there is nobody else available to light the candles, then the "Onen" should light without reciting a Beracha. This is the ruling of the Kaf Ha’haim (Orah Haim 470:20).
One should not visit a cemetery during Hanukah, even on the anniversary of a parent’s passing. If one’s parent passed away during Hanukah, he should visit the gravesite before Hanukah, rather than during the holiday. It is permissible, however, to pray at the graves of Sadikim during Hanukah.
The Kaf Ha’haim (Orah Haim 470:26) records a custom to recite seven times after lighting the Hanukah candles the verse of "Vi’hi No’am" (Tehilim 90:17) followed by the chapter of "Yosheb Be’seter" (Tehilim 91). He writes that this recitation is a Segula ("charm") for protection from harm.
