According to Red Dwarf, deathday is the anniversary of the day a hologram died. Living people celebrate birthdays, dead people celebrate deathdays. The very first deathday celebration in Red Dwarf is held in the first episode, The End, for George MacIntyre which they held right after his funeral.

You also get to see Rimmer's deathday celebration in Red Dwarf in the episode Thanks for the Memory, when the crew hang out on a small moon playing air guitar and getting drunk. It is the wee hours of the morning following his celebration when Rimmer explains to Lister he's only had sex once, with Yvonne McGruder. Lister sympathizes with Rimmer, and decides to give him Lisa Yates as a present.

See also gazpacho soup.