An article by Nanna Rognvaldardottir, an Icelandic food expert, states that the drink adopted the nog part of its name from the word noggin, a Middle English phrase used to describe a small, wooden, carved mug used to serve alcohol in. Another name for this English drink was Egg Flip. Yet another story is that the term derived from the name egg-and-grog, a common Colonial term used to describe rum. Eventually the term was shortened to egg'n'grog, then eggnog. In 1607 reported by Captain John Smith. Egg Nog was first consumed in what would become the future United States in Jamestown, Virginia.
Grog was half and half rum and water. Dunno if the water or the rum masked the taste of the other ---
If I recall my naval history, Grog was indeed a way to make stale, old water more drinkable. Recall that sail ships could only carry water in casks and barrels, and that water would get rancid (slimy and yellowed by one account). Adding the rum to make grog not only allowed the rancid water to be drank, but it also stretched the rum allotment, the tot, each sailor was entitled to.
