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
This was on my list of questions to ask at this site! I don't have an answer but I do have some thoughts and further questions.
Most of these words seem to be variants.
If you look at the list here at Words Shakespeare invented, most of these are variations. For example amazement is a variant of amaze. It’s a little more credible to vary an existing word than introduce a completely new one. Also could it be true that there was a lot more improvisation in the language generally at the time due to the lack of a literate populace and established written standards and dictionaries.
First to use in print isn't the same as invent
If this was the dawn of English language publishing then presumably Shakespeare was just in the right place at the right time for be the first to print with many English Words.
What of Shakespeare's Rivals?
If Shakespeare was in the right place at the right time then so surely were Marlowe and Jonson - I haven't seen similar claims for them.
Does every language have its Shakespeare?
I've heard Luther being called the father of German. Would Homer be the same for Ancent Greek? What about other languages?
Best Answer
You have noticed a very peculiar aspect of English vocabulary. As rich as it is in comparison to many other languages, due to its almost creole history, it really is impoverished in comparison to other languages in kinship terms.
But 'why' is always a difficult question, especially when mixed with cultural questions. There are the difficulties with Sapir-Whorf type explanations: both language restricting thought on one hand and the number of Eskimo words for snow on the other.
Does the lack of kinship terms reflect the cultural lack of warmth and caring for relatives among English speakers, that is not caring leads to the loss of the words (which etymologically do exist in the ancestor languages), or did the arbitrary lack of kinship terms contribute to the crumbling of English family values?
Any direction sounds much too tendentious, too judgmental, and requires too much unjustified and biased assumptions to choose.
The comparative lack of kinship terms does ask for an explanation but one backed up by linguistic and anthropological and comparative research. The only source that comes to mind is Levi-Strauss's 'The Elementary Structures of Kinship.' (primarily anthropological but as a by product a number of examples of kinship term systems.
English isn't alone in having relatively few kinship terms. Some other European languages have only a few extra (French, German) and some languages really only have names for their clan and generation (anybody of one's biological parents' generation might be called something like 'uncle' or 'aunt', even one's birth parents).
Having no definite answer to your question, I can only say beware of making cultural inferences based on restrictions to languages. Some languages have grammatical gender and others don't, but that doesn't mean the ones without can't recognize the sex of other people.