Originally, Rudolph's name is never mentioned specifically because of copyright reasons. Only because of this was its omission then worked into the plot as a running gag.
As Robbie walks out into a blizzard, thinking he'll be a burden to the team, he says, "I'm going outside. I may be some time." This is a paraphrase of Captain Oates's last words of the ill-fated Terra Nova expedition in 1912. Oates, suffering from gangrene and frostbite, left his tent and walked outside into the Antarctic cold, and died of hypothermia. This is seen as an act of self-sacrifice so his team could move faster toward rescue.