Being of the curious sort I decided to chase this down the ole rabbit hole.
I always like to start in ancient Egypt and seeing that the stork is indigenous to the African continent, I was confident the world's greatest natural philosophers could help me out with this question. The hieroglyph for the ancient Egyptian concept of our immortal soul, the ba, is the saddle-billed stork.
Due to our Judeo-Christian upbringing, we would think of this most unfortunate result as being sent to eternal damnation and Egyptologists will describe the outcome as such. It's not an apt description of what is being depicted here; however, if you never change then I guess there is some truth to that. I should also touch on the concept of the ka. The ka is your manifestation, the life-creating force, on this plane of existence. The ancient Egyptians described not only humans, but animals, plants, water and minerals as having a ka and the hieroglyph for ka is a bull - a bull being a great symbol for the dynamism of material existence. You will have many kas before your journey is over, but only one ba. Your ba makes each ka and is the vessel that carries your multitude of kas. The ancient Egyptians symbolized this with the ram god Khnum, who was known to fashion humans on his potter's wheel.
After your ba has made a new ka, this material life force is now able to house its ba. To incarnate into matter once again is a perilous journey for the ba, and is mythologized in the story of Osiris being hacked to pieces and scattered throughout Egypt by his nemesis Set. Isis, after much weeping, then embarks on a quest to retrieve the remains of Osiris and puts him back together in order to engender a new life, the offspring known as Horus. In this reconstituted form, Osiris' mummy is depicted wrapped in a fine single strip of white linen. Also of note is the child known as Khonsu who is closely associated with Horus. Khonsu, as a child of Amun and Mut, represented the child waiting to be born inside of us - in this case the mother's womb known as Mut. In this aspect, Khonsu would be depicted as a mummiform child, wrapped in linen, awaiting birth to take his rightful place on the throne of man as Horus. Khonsu, in the pyramid texts, is referred to as the one who lives on hearts. This is generally thought of as a bloodthirsty aspect of Khonsu; however, it is really telling us that it is the heart that will ultimately feed and quicken the birth of the enlightened soul in us.
Alright, so, I'm thinking that the word baby might have some connection to the ba of ancient Egypt due to the hieroglyphic for ba which is the stork and the way the baby is delivered by the stork wrapped in linen. However the word baby does not have a clear history or meaning that correlates in this regard. Our old friend, the Online Etymological Dictionary tells us to look up the word "babe" instead and reveals this of the word:
late 14c., short for baban (early 13c.), which probably is imitative of baby talk (cf. babble), however in many languages the cognate word means "old woman" (cf. Rus. babushka "grandmother," from baba "peasant woman").
Slightly amusing is that the word "babe" can be traced in many languages as the word for an old woman or a peasant woman while nowadays we call an attractive women a babe! There is also the word "body" which has intrigued me for awhile. The Online Etymological Dictionary reveals this of the word:
Old English bodig "trunk, chest" (of a man or animal); related to Old High German botah, of unknown origin.
This is interesting because our body is a container for the immortal soul called the ba. The old word botah has an ending "tah" that is a common ending when describing something that has material form such as a place or land. In ancient Egypt, land was feminine and given the half-circle hieroglyphic which is phonetically rendered as "ta". Also, well known is the transpositional nature of the letters d and t throughout the history of languages, which accounts for botah becoming body. I'm willing to connect the ba with the ta being our body, though I'm sure it could be considered a stretch. Given the evidence I've presented above, I think it's plausible. As well, the connection of baby to a peasant woman can't be ignored since it is the woman as Isis within us, who is the womb of the ba which has incarnated into matter. Everyone's ba is associated with this woman waiting to be born.
Another tantalizing thing to consider is the connection of the word babe to an old woman. I bet you don't have to get too far into the Old Testament and the story of Abraham's wife Sarah before you start asking yourself some serious questions about where the myths come from. Sarah was 90 years old when she first conceived.