baby:  Like mama and papa, baby and the contemporaneous babe are probably imitative of the burbling noises made by an infant that has not yet learned to talk. In Old English, the term for what we would now call a ‘baby’ was child, and it seems only to have been from about the 11th century that child began to extend its range to the slightly more mature age which it now covers. Then when the word baby came into the language, it was used synonymously with this developed sense of child, and only gradually came to refer to infants not yet capable of speech or walking.
late 14c., babi, diminutive of baban (see babe + -y (3)). Meaning "childish adult person" is from c. 1600. Meaning "youngest of a group" is from 1897. As a term of endearment for one's lover it is attested perhaps as early as 1839, certainly by 1901; its popularity perhaps boosted by baby vamp "a popular girl," student slang from c. 1922. As an adjective, by 1750.
Baby food is from 1833. Baby blues for "blue eyes" recorded by 1892 (the phrase also was used for "postpartum depression" 1950s-60s). To empty the baby out with the bath (water) is first recorded 1909 in G.B. Shaw (compare German das Kind mit dem Bade ausschütten). Baby's breath (noted for sweet smell, which also was supposed to attract cats) as a type of flower is from 1897. French bébé (19c.) is from English.