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
The term dad has origins in children's speech:
recorded from c.1500, but probably much older, from child's speech, nearly universal and probably prehistoric (cf. Welsh tad, Ir. daid, Czech, L., Gk. tata, Lith. tete, Skt. tatah all of the same meaning)
Daddy is the diminutive of this:
c.1500, colloquial dim. of dad, with -y
The OED quoted here adds more, saying:
Occurs from the 16th c. (or possibly 15th c.), in representations of rustic, humble, or childish speech, in which it may of course have been in use much earlier, though it is not given in the Promptorium or Catholicon, where words of this class occur.
Of the actual origin we have no evidence: but the forms dada, tata, meaning 'father', originating in infantile or childish speech, occur independently in many languages. It has been assumed that our word is taken from Welsh tad, mutated dad, but this is very doubtful; the Welsh is itself merely a word of the same class, which has displaced the original Celtic word for 'father' = Ir. athair.
A childish or familiar word for father: originally ranking with mam for mother, but now less typically childish. Cf. daddy.
?a1500 Chester Pl. (Shaks. Soc.) I. 43 Cayme. I will..Speake with my dadde and mam also..Mamme and dadd, reste you well! [Of uncertain date: the MS. is only of 1592. Harl. MS. of 1607 reads (ii. 678) 'sire and dam', (ii. 681) 'father and mother'.]
1553 Wilson Rhet. 31 Bryngyng forthe a faire child unto you..suche a one as shall call you dad with his swete lispyng wordes.
So dad or daddy stems from baby talk. This makes sense—f is a difficult sound for babies to say, but "harder" sounds like d are easier. Note, however, that just because many cultures share the same-sounding word (dada, tad, tata), this does not mean that one can make a clear distinction of origin. Though tata means dad, it does not provide substantial evidence that dad is from tata. As the OED points out, the form occurred independently in many languages.
Best Answer
In this instance, the speaker is mocking the speech patterns of some people from New York and New Jersey. The accent that is mocked is not prestigious; it is associated with a lack of education and some crudeness. Priests are well-educated, so it is humorous that this priest is using a "lower-class" accent.
Using the "d" sound instead of "th" is widespread. "Who dat?" is almost a slogan of New Orleans, home of many Creole speakers. French people speaking English often have difficulty with "th", "d" is how they approximate the sound. It is also a speech characteristic of some Blacks from the southern United States and English people from Yorkshire.
The Alan Sherman song, "Hello Muddah, Hello Faddah" derives a lot of its humor from the contrast of this lower-class accent with the classical music used for the melody.