On my last cruise - aboard the Oasis Of The Seas - my bartender friend Reynald made me his "special" rumrunner. He squeezed juices from limes and oranges (no mixes or pre-made mixers), carefully added various rums and other things, and it was really the best cocktail I've ever drank.
I guess if I had to pick just one it would be a BBC made with real bananas. Like Dave, I've made friends with a few ship bartenders over the years and they'll make drinks for me from fresh ingredients sometimes. I don't usually drink frozen drinks very often..but man, that hit the spot after walking all day in the heat.
It may not be classified as a "drink", but I always enjoy a hot cup of coffee with hot chocolate mixed in every morning on a cruise.
RCI Radiance of the Seas Sep 2009
NCL Sun March 2008
Carnival Destiny June 2006
Carnival Ecstasy June 2002
Carnival Elation August 2000
Celebrity Century June 1998
Carnival Carnivale August 1993