Many believe that eggnog is a tradition that was brought to America from Europe. This is partially true. Eggnog is related to various milk and wine punches that had been concocted long ago in the "Old World". However, in America a new twist was put on the theme. Rum was used in the place of wine. In Colonial America, rum was commonly called "grog", so the name eggnog is likely derived from the very descriptive term for this drink, "egg-and-grog", which corrupted to egg'n'grog and soon to eggnog. At least this is one version...
Other experts would have it that the "nog" of eggnog comes from the word "noggin". A noggin was a small, wooden, carved mug. It was used to serve drinks at table in taverns (while drinks beside the fire were served in tankards). It is thought that eggnog started out as a mixture of Spanish "Sherry" and milk. The English called this concoction "Dry sack posset". It is very easy to see how an egg drink in a noggin could become eggnog.
The true story might be a mixture of the two and eggnog was originally called "egg and grog in a noggin". This was a term that required shortening if ever there was one.
With it's European roots and the availability of the ingredients, eggnog soon became a popular wintertime drink throughout Colonial America. It had much to recommend it; it was rich, spicy, and alcoholic.