The questioner asks where the word ‘dad’ comes from as used by children.
Surely it and ‘mum’ arise as they are two sounds easily made by babies.
You know dad dad dad dad , mum mum mum mmm etc as they practise making sounds. The proud parents associate these sounds as being directed to them as they stare lovingly at their offspring.
Judging from my own experience, it’s a miracle that sound didn’t come to mean “big sister”. In point of fact, my first intelligible uttering was “big sister”, or a reasonable approximation thereof.
Granted, a rather WAG,and though I’ve searched, can’t find substance, so will go with the guess. MaMa is a nursing cry, mimicing the mouth at the breast. DaDa is referring to the Other guy, over there, Da …for lack of a better term.
Pa might be that sort of infantile directive, too.
I remember reading (and I’m unable to cite, of course), that it’s easier for infants to make an open-mouth sound first, and then close their lips, so the sound produced would be “amma” or “abba.” Abba being, of course, the word for “father” in Hebrew and Greek. I could be wrong about the languages, and I’m open to correction. The gist of the piece was that in those patriarchal societies, they made sure that the first sound an infant produced would be the word “father.” I’m sure this is conjecture.
Except that the various “m” sounds (all of which mean mother) generally come before the “d”, “b”, or “p” sounds (which mean father, in various languages and dialects).