For a lot of modern pagans, this is the time of year when it is especially appropriate to honor the dead. The Irish Samhain, widely observed in the modern pagan community, the heathen Winter Nights, even the Christian All Souls Day and secular Halloween are reminders that the dead are still with us, albeit in their own way and (usually) in their own place.
For this reason, some Hellenic pagans choose to observe the Genesia at this time; although it was held on 4 Boedromion which would more properly place it in mid-September, that’s not an unreasonable choice. The precise nature of the historical Genesia is unknown, though there may have been both a public (civic) aspect and a private aspect to it, as was the case with some other festivals. The link between the Genesia and the practice of families honoring their own dead on their individual anniversaries seems an obvious one. It would certainly have been simpler to honor all of one’s dead during a single festival than to keep track of all the anniversary dates. The Genesia was also an occasion to pay honor to Ge or Gaia, within whom the dead made their home.
Another Athenian festival of the dead takes place during the Anthesteria, a three-day festival held on 11 Anthesterion (approximately 19 February) in which the first two days were dedicated to Dionysos as god of the new wine while the third honored Hermes Chthonios (Hermes as psychopomp or guide of the dead) and paid homage to the dead. This day seems to have had a more strictly apotropaic, more fearful tone than the Genesia.
In Erchia and elsewhere the dead were honored as the Tritopatores (“great-grandfathers”), a term for the ancestors as a group.
The relationship of the ancient Greeks with their dead was a complex one. On the one hand, honoring the dead was a sign of love, a way to show your devotion to those who had come before you. On the other hand, it was also a propitiatory act, a way to keep potentially-dangerous spirits from harming you or your living family. Modern pagans will likely focus primarily on the former aspect, but the latter should not be forgotten because it is a reminder of the power held by those who have passed into the next world. It need not be a matter of solely one or the other.
Traditionally, offerings to the dead were poured out onto the ground as was usual for chthonic entities. These offerings could be of various sorts: milk, honey, wine, olive oil, barley broth, even water was appropriate.
(Note: While wine is sometimes not recommended as an offering for chthonic spirits, milder drinks being preferred, it is also attested as an offering to the dead. My own sense–and this is UPG, I can’t support it solidly via the sources so far–is that while wine might be inappropriate for an offering meant for solely or primarily propitiatory purposes, it is probably fine for an offering given simply in love and devotion, especially if your ancestors were in fact fond of wine. Personally I might give offerings of coffee; since there was always a pot brewing at my grandma’s house I know it would be a welcome gift.)