According to World Wide Words, it originates from firemen doing speed competitions without carrying water.
The term run, more fully fire run, has for at least the past century been used by local fire departments in the USA for a call-out to the site of a fire. It was once common for fire departments or volunteer hose companies to give exhibitions of their prowess at carnivals or similar events. [...] These competitions had fairly standard rules, of which several examples appear in the press of this period, such as in the Olean Democrat of 2 August 1888: “Not less than fifteen or more than seventeen men to each company. Dry run, standing start, each team to be allowed one trial; cart to carry 350 feet of hose in 50 foot lengths ...”.
These reports show that a dry run in the jargon of the fire service at this period was one that didn’t involve the use of water, as opposed to a wet run that did. In some competitions there was a specific class for the latter, one of which was reported in the Salem Daily News for 6 July 1896: “The wet run was made by the Fulton hook and ladder company and the Deluge hose company. The run was made east in Main street to Fawcett’s store where the ladders were raised to the top of the building. The hose company attached [its] hose to a fire plug and ascending the ladder gave a fine exhibition.”
It’s clear that the idea of a dry run being a rehearsal would very readily follow from the jargon usage, though it first appears in print only much later.
Andrew Leach's answer has the OED's first quotations [parenthetically in 1884, and] in 1890. Their first quotation for "in good nick" is The English dialect dictionary from 1905.
Australia, 1880s
I found earlier uses in the Trove archive of Australian newspapers, the earliest in The Referee (Sydney, NSW, Thursday 13 January 1887):
Hutchens and Samuels.
(By "Shoespike.")
Next Monday Hutchens will run his first
match in Australia. Malone's was to have
been the first, but the aboriginal party
were found willing to risk a century, and
a match was quickly made. Samuels has
not had much time for preparation, but is
quietly doing work on the Agricultural
Ground. He looks if anything fine, and
not so strong and in such good "nick" as
when he won the Botany. As an aboriginal
Samuels is a first-rate runner, and about
the best of them. I question, however, if
he is class enought to stretch the world's
champion and anticipate Hutchens to
win comfortably. I may add I do not
expect even time to be broke.
It was used in other Australian newspapers in the late-1880s to describe sporting participants: wrestlers, racehorses, footballers, boxers rowers.
New Zealand, 1870s
However, it can be found earlier in New Zealand's archive of newspapers, Papers Past, and again in a sporting context. First in Sporting Notes by "Sinbad" in The Press (Volume XXIX, Issue 3973, 18 April 1878, Page 3), describing racehorses:
York, the representative of the Bay stable, is big enough and strong enough. Those who ought to know say he has plenty of pace, and will certainly be there or thereabouts at the finish. He is without doubt in good nick, and will have a good man on his back, so I think he will run into a place, and if either Natator or Merlin are out of it he may be labelled dangerous.
(The article also uses the similar phrase in good form.) In good nick shows up in many other editions of The Press and also The Obago Witness in the late-1870s, all applied to racehorses.
An origin?
Another meaning of the noun nick dates from 1824 and, according to the OED:
10. An instance of cross-breeding, esp. one which produces offspring of high quality. Cf. nick v.2 7b.
You could say of animals or racehorses, as in this from an 1870 Australian newspaper:
It is possible,
however, as the mare is a daughter of Melbourne, that
Stockowner may prove a good nick.
From the same article, as a verb:
I see that a certain sire and dam "nick" well, no
matter how wrong it may be for them to do so, as far as
the relationship of their families is concerned, I prefer
to trust to their progeny, rather than to thoso bred on a
correct theory without practical results.
So perhaps as the term for successfully crossed animals, specifically racehorses, was applied to racehorses generally in good form. This was then used for sportsmen in general before being used for anything in good condition, or conversely, as "in poor nick" for something in bad condition or form.
Best Answer
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