Feelings of depression or anxiety, experienced by some mothers following childbirth.
Had anyone mentioned 'baby blues' prior to WWII they would have been thought to have been talking about colour - specifically the colour of someone's eyes. Most babies are born with blue eyes due to a lack of the melanin pigment until sometime after birth. The use of the term 'baby blues' to mean eyes is a natural development, which came about in the USA in the early 20th century; for example, this from the American author Rex Ellingwood Beach's novel Winds of Chance, 1918:
"Fix your baby blues on the little ball and watch me close."
In the 1940s 'baby blues' began to be used with the meaning we now usually give it, i.e. post-natal depression.
In his best-selling baby care book Expectant Motherhood, 1940, Nicholson J. Eastman wrote:
"Most common among such reactions, perhaps, is what is colloquially called the 'Baby Blues'."
Interestingly, in a later 1960s edition of the book Eastman suggested that expectant mothers should limit themselves to no more than ten cigarettes per day. How times change.
I have found an earlier mention of 'baby blues' in a 1909 edition of The Syracuse Herald:
"There are various kinds of blues - navy blues, afflicting those who view Captain Hobson with alarm; baby blues, following the second pair of twins, etc."
This is a jocular piece which lists various plays on the word blues. I don't think it can be viewed as an early citation of the 'baby blues' phrase with its current meaning, more a coincidence.