I extracted the following from the online Webster Dictionary. It's interesting to see how all these words were transformed from Latin/Greek/Old High German/Middle English to the current English words.
Note: I know that this does not exactly answer your question, since you actually want to know if the words derived from words with other meanings. But I think that having the full list of originating languages here may be useful as other answers to your question may refer to it.
DAUGHTER
Middle English, doughter, from Old English dohtor; akin to Old High German tohter daughter, Greek thygatēr
First Known Use: before 12th century
SON
Middle English sone, from Old English sunu; akin to Old High German sun son, Greek hyios
First Known Use: before 12th century
AUNT
Middle English, from Old French ante, from Latin amita; akin to Old High German amma mother, nurse, Greek amma nurse
First Known Use: 14th century
UNCLE
Middle English, from Anglo-French, from Latin avunculus mother's brother; akin to Old English ēam uncle, Welsh ewythr, Latin avus grandfather
First Known Use: 14th century
MOTHER
Middle English moder, from Old English mōdor; akin to Old High German muoter mother, Latin mater, Greek mētēr, Sanskrit mātṛ
First Known Use: before 12th century
FATHER
Middle English fader, from Old English fæder; akin to Old High German fater father, Latin pater, Greek patēr
First Known Use: before 12th century
COUSIN
Middle English cosin, from Anglo-French cusin, cosin, from Latin consobrinus, from com- + sobrinus second cousin, from soror sister — more at sister
First Known Use: 13th century
NEPHEW
Middle English nevew, from Anglo-French nevou, neveu, from Latin nepot-, nepos grandson, nephew; akin to Old English nefa grandson, nephew, Sanskrit napāt grandson
First Known Use: 14th century
NIECE
Middle English nece granddaughter, niece, from Anglo-French nece, niece, from Late Latin neptia, from Latin neptis; akin to Latin nepot-, nepos grandson, nephew
First Known Use: 14th century
Reference:
http://www.merriam-webster.com
Best Answer
According to wiktionary:
From Middle English moder, from Old English mōdor, from Proto-Germanic *mōdēr (cf. East Frisian muur, Dutch moeder, German Mutter), from Proto-Indo-European *méh₂tēr (cf. Irish máthair, Tocharian A mācar, B mācer, Lithuanian mótė).
That's abosultely right. Proto-Indo-European is the hypothetical ancestor language or protolanguage of most European and Indian languages.
That's why in many languages of the same origin the word "Mother" is used with trivial variations. I'm don't have a listing of the words you're looking for.
Note that some words might have been used in other languages because of reasons other than language origins. For example many Arabic words are used by Muslims in middle east in countries like Turkey, Pakistan, Iran, etc. Or some other words like okay are gaining popularity in different languages and get used by many people. But as RegDwight mentions for the word okay this is a case of borrowing a word.
The word Mama or Papa are one of the easiest words that can be produced or repeated or by babies. Maybe that's one of other reasons which has made the words being used in most of the languages around the world.
To get more information about Proto-Indo-European language visit here.
To get more information about the list of Proto-Indo-European languages visit here.