The word nog comes from a toast given in old English. Originally, the toast was given with an ancient form of beer. From there, it was adopted to an uppercrust English dairy creation, eggnog, which was milk or cream mixed with eggs and brandy, in the early 19th Century, also known as an egg flip and probably decended from a hot beverage called "posset", not to be confused with the Brooklyn egg cream.
The use of rum appears to be an American invention with early trade with the Caribbean islands, since rum was cheaper than brandy, sherry or other taxed spirits from Europe. Regional variations abound, like the use of burbon in Kentucky and throughout the south. Germans still make a version laced with beer.