When we first met Kate, our adoptive Chinese daughter, at 13 months old, she was already babbling, saying "mama(-m)" and "baba(-b)". Surprisingly to me, she seemed to use them differentially -- "mamama" when she was upset or needing something, "bababa" in a playful mood. I had noticed the same thing in my son when he was about 10 months old, although for him the second sort of babbling sounded like "dadada". At the time, I figured Davy's differentiated babbling was due to our parental reinforcements and interpretations -- that when he said "mama" we were more likely to hand him to his mom, and when he said "dada" we were more liked to hand him to dad, and the like. What was striking to me was that, as an orphan, Kate had no mom or dad. (It's not even clear that she had ever seen a man before.)
Since at least the 1960s, linguists have known that a wide range (maybe even of majority) of languages of very different origins use something like the "mama" (or "nana") sound for mother and something like the papa/baba/tata/dada sounds (all related phonologically) for father. For example, in Chinese "mama" and "baba" are baby-ese for mother and father, in Hebrew it's "ima" and "abba", and of course there's Spanish, Italian, etc.
The standard first remark about this is that these words are very easy for babies to say. I wonder if in addition to this, "mamama" goes more naturally with need or distress while the other goes more naturally with fun and exploration -- for example if the "m" phoneme is easier to make with the facial expression of distress (it seems to me that m's are easier with a tight, distressed face, p's and d's easier with an open, relaxed face, but maybe that's just me). That would explain in a way other theories would not why even an orphan would show the sort of differentiation Kate does between those two sounds. Given the different roles mothers and fathers typically play, it's then easy to see why one babble would come to be associated with the mother and the other with the dad.
(Probably some linguist has suggested this before, but for obvious reasons I haven't time right now to do serious reference chasing. Even without the reference chasing, though, I feel comfortable doubting whether anyone has thought to study 12-month-old orphans specifically in this connection.)