What do they do when somebody (e.g. an elderly passenger) dies aboard a cruise ship?

depends where the ship is. Either keep the body in the morgue most cruise ships have, or do an unscheduled stop to disembark the body and any family members - or if they are bound to a stop anyway, offload there. (of course if further transportation is possible)

Each oceangoing cruise ship is required to carry body bags and maintain a morgue. Separate from food storage areas, most morgues are small, with room for three to six bodies.

On standard Caribbean sailings, remains are typically kept in the onboard morgue until the ship returns to the United States where a death certificate can be issued by the local medical examiner's office, de la Cruz said.

But when the ship is far from its homeport or doesn't have a homeport, the body must be repatriated from somewhere.

http://www.cruisecritic.com/articles.cfm?ID=1631

Carnival Imagination bent over backwards to help the family when my son-in-law's father died shortly after we went through the Panama Canal. They gave him free internet, phone, and all the photos that had been taken of everyone in the family group. They got in touch with funeral home people in Long Beach (our final destination) and insurance people. Everything was very well and tactfully taken care of.

First thing they do is go through their room and dig out anything valuable. Then they dress the old people up in swimsuits and go sit them out on the deck of the ship on beach chairs. Slather them up with enough sun block, and nobody notices there's a dead rotting corpse lying there.

they are placed in the ship's morgue until they return to home port

Enjoy your trip

Chuck them overboard