Most languages probably have a few words used especially for addressing babies. However, Siwi seems to have a lot more than I know from English or Arabic (I've recorded something like 40). One of these (already noted in Laoust 1931) is mbuwwa "water" (the normal Siwi word is aman). mbuwwa, meaning "water" or "drink", turns out to be rather widespread: they use it in baby talk in Syria, Lebanon, Tunisia, Algeria, Morocco, Malta, Sicily, and probably a few other places for which I haven't found sources. The remarkable part is that Ferguson managed to track down a historical source for this word. Varro, a Roman grammarian of the first century BC, gives bua as the nursery word for "drink" (presumably to be related to bibere, the adult verb for "drink".) (Unfortunately, I haven't managed to find the relevant work online.) If the connection is correct, then this word (possibly along with some others, like pappa for "bread" or "food") has persisted in Mediterranean baby talk for at least 2000 years, apparently without ever passing into adult speech.
So what special words do you use in your language when talking to babies?