late 14c., babi, "infant of either sex," diminutive of babe (q.v.) with -y (3).
The meaning "childish adult person" is from c. 1600. The sense of "youngest of a group" is by 1897. As a term of endearment for one's lover it is attested perhaps as early as 1839, certainly by 1901 (OED writes, "the degree of slanginess in the nineteenth-century examples is not easily determinable"); its popularity perhaps was boosted by baby vamp "a popular girl" (see vamp (n.2)), student slang from c. 1922.
The meaning "minute reflection of oneself seen in another's eyes" is from 1590s (compare pupil (n.2)). As an adjective by 1750. Baby food is from 1833. Baby blues for "blue eyes" recorded by 1892, perhaps for the reduplication as well as the fact that more infants have blue eyes than keep the color (the phrase also was used for "postpartum depression" 1950s-60s).
To empty the baby out with the bath (water) is attested by 1909 (in G.B. Shaw; compare German das Kind mit dem Bade ausschütten, attested from 17c.). A baby's breath was noted for sweet smell, which also was supposed to attract cats, hence baby's breath as the name of a type of flower, attested from 1897.
French bébé (19c.) is said to be from English, but there were similar words in the same sense in French dialects.