Origins of the word “mom” and "mother"

Solution 1:

"mama" is a pre-verbal vocalisation by children - usually one of the first sounds that they make. Often infants will say "mamamamama" as one of their first vocalisations, and this often doesn't appear to have any real meaning - it's not clear that they are addressing their mothers or if they are just practising speaking. I don't know why this particular sound emerges, but I assume it's an accident (ie biological phenomenon) of the way that the brain and speaking organs function or develop. It might be the easiest speech sound to make.

So, I suspect that "mama" has entered the language because young children naturally say it, rather than children saying it because it's in the language.

Later on in the child's development, "mama" properly takes on the meaning of "mother", ie the child deliberately says "mama" to attract their mother's attention for example. This might be because the mother says it back to them, or there might be something about the word itself which is intrinsically linked to the mother, in the child's mind: perhaps it always has some degree of semantic function, in a primitive way.

Further reading:

http://www.livescience.com/32191-why-are-mama-and-dada-a-babys-first-words.html

http://theweek.com/articles/464678/why-babies-every-country-earth-say-mama

Solution 2:

As Hot Licks and Max Williams correctly pointed out, mama is probably the simplest sequence for a baby to say. But nobody has yet pointed out the connection between mama and mammary, from Latin mamma - the breast. And a baby at the breast can be heard making mama sounds while suckling.
But that doesn't mean that to the baby the sounds mean either mother or breast. Those associations are learned later from adults.

In Gujarati, mother is ba for Hindus, whereas for Muslims, father is baji. Mama is mother's brother, and mami is his wife. So there are as many differences as similarities.
As for Chinese, ma has 5 different meanings, according to the tones used.
Mama, dada, nana, baba, papa are all very simple sequences you will hear in the babbling of all babies everywhere, so it's not strange that all of them occur as relationship terms in many languages. They're part of human speech - the ability to produce sounds - which is innate.
Human language - the ability to use sounds to convey meaning - is not innate, and must be learned. Sometimes different groups of humans assign the same meaning to the same sound sequence, just by accident.

Only when you find a statistically significant set of such parallels should you begin to suspect a relationship between the languages.