It has been observed that in general, a word for "mother" tends to be based on a bilabial nasal [m] or similar consonant, and for father it tends to be [b] or [p]. This is found in many language families, so they can't be considered cognates. This is usually explained by the easiness of babies making this sound early in their babbling, so the parents just use those sounds to refer to themselves. However why is it that few languages have used the reverse? (i.e. [p]/[b] for mother and [m] for father)? Wikipedia lists only Georgian, where "father" is /mama/ and "mother" is /deda/.
This issue is discussed in some detail in "Where do mama/papa words come from?" by the late, great linguist Larry Trask. The paper also gives a very nice introduction to argumentation in historical linguistics.
The answer is that these terms are based on the earliest 'intelligible' babble of babies. The most common first syllable produced by babies is [ma], with [pa/ba/ta/da] following soon after. These earliest articulations are probably just play for the child, but are interpreted by parents as attempts by the baby to address them. As mothers tend to be the main early caregiver the earliest-occurring syllable, typically [ma], is interpreted as referring to them, while the next-occurring syllables are very commonly interpreted as referring to the father. This gives rise to the strong, but not invariant, association of [ma] with 'mother' and [pa/ba/ta/da] with 'father'.
Although it is a strong tendency it is still only a tendency. The prevalence of these two sounds in the names of parents is not surprising, given that they're two of the easiest sounds to make regardless of the sound system of your language. Think about English - we have 'mother' and 'father' but we'll accept 'mama' or 'dada' as a first word, because they're easy to make - open CV structure and basic stop sounds.
Basically, across the world caregivers have an incentive to hear what they want to hear - and they want to hear their children say their names. Given that cross-linguistically mothers are generally caregivers, the easiest sound 'ma' usually is used first and the harder sound 'pa' or 'da' somewhere around second. Children are already very aware of who their caregivers are before they can articulate that, so as soon as they can articulate something, even something as simple as 'ma' this gets attributed as a name.
So it's less that these words are a strong cross-linguistic tendency and more that parents are anxious to hear meaning in a child's early babbling-like word production!
Let me add to these already great answers what I read in a paper (I cannot recall, which): m is a nasal sound that can be produced while suckling on the breast, and suckling involves both lips (hence m is a bilabial nasal). I am not sure if the argument goes "They can utter the word for 'mom' while suckling" (unlikely) or "They associate the movement and the sound they produce while suckling with the person". Anybody came across this paper?
sorry if this had yet been answered sufficiently
a) mama variants are significantly more common than papa (so I heared). Indeed, p is learned far later (else where it's said to be "soon after") and thus deemed more difficult than m. Therefore let's focus on mama. By the way, you'll find a similar distribution of [m]e in translations of the first person singular pronoun.
b) You wrote: "This is found in many language families, so they can't be considered cognates" -- that's presupposing that the language families aren't releated, but that is as of yet an open question. This topic and your argument (see c below) specifically is argued to deny a common ancestor of the world languages, so you kind of have a circular argument.
c) "This is usually explained by the easiness of babies making this sound early in their babbling, so the parents just use those sounds to refer to themselves" -- I have read the argument (and will find the news article about the prof., ca. from the time this Q was posted), that it relates to the lip motion of sucking on tiets. I conclude that does at least explain why the phoneme is easy to produce (although it's prerequisite vice versa for eating, that the motion is possible). I'd further inferred from the article that the motion is liable to be produced when hungry (or when satisfied, hmmmmm). In that sense, mama et al might be phonosemantic and especially priviliged to survive. If phonosemantics is not your thing, one could at least argue if the sound is so likely to be primed for mum that it would be likely to survive if there was a common ancestor and more likely if reinforced by cultural background.
Note that while the sound may be easy to learn, it may be not as easy to recognize as an appelation. At least to "just use those sounds to refer to themselves" is a gross simplification.
1: I didn't find the news article, but the claim can be found by Roman Jakobson, Why 'Mama' and 'Papa'?