Wikipedia:
Most writing on the egg cream assumes that it originated in New York City and most often that it originated among Eastern European Jewish immigrants. This has led to a variety of claims meant to explain the widely noted paradox that the New York City egg cream is made with neither eggs nor cream.
One theory is that grade “A” milk was used in its creation, leading to the name “a chocolate A cream”, thus sounding like ‘egg’ cream. Stanley Auster, the grandson of the beverage’s alleged inventor, has been quoted as saying that the origins of the name are lost in time. One commonly accepted origin is that “Egg” is a corruption of the German (also found in Yiddish) word echt (“genuine” or “real”) and this was a “good cream”.
Food historian Andrew Smith writes: “During the 1880s, a popular specialty was made with chocolate syrup, cream, and raw eggs mixed into soda water, In poorer neighborhoods, a less expensive version of this treat was created, called the Egg Cream (made without the eggs or cream).”
Darcy S. O’Neil, author of the book Fix the Pumps, a historical look at soda fountains, claims that the “New York Egg Cream” is a variation of the original milkshake served at soda fountains throughout America in the late 19th century.
Around 1885 the milkshake became a popular item at soda fountains. Unlike today’s thick, ice cream-like consistency, the original milkshakes were made with sweet cream (sometimes frozen as “ice cream”), a whole egg, flavored syrup, and soda water. The egg, cream, and syrup were shaken in a cocktail shaker until light and frothy, then poured into a glass where the soda water was added.