Where does the phrase "in good nick" come from?
Solution 1:
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.
Solution 2:
OED doesn't seem to know. It lists this use in classification IV., "Other uses":
16. colloq. Condition, state. Chiefly in in good (fair, etc.) nick : in the specified state or condition.
[1884 R. Lawson Upton-on-Severn Words 39 ‘Up to dick’, or ‘nick’..= in first-rate condition; to perfection.]
1890 J. D. Robertson Gloss. Words County of Gloucester 103 Nick,..condition, fettle.
...which would indicate it's a West Country dialect word. As such, it's almost certainly a lot earlier than 1880 — just it was first recorded around then.
Swindon, a major railway town, is not all that far from Gloucester, and it's possible that "in good nick" spread from the West Country with the railway, which would also tie in with the first recording of the use.
OED notes the word first appeared in Middle English, but they have no certain etymology.
Etymology: Origin unknown; compare nick v.2 Several of the senses of the noun (in branches I.) correspond closely to senses of the verb (compare branches I. and II. s.v.), but are older; the noun may in reality have priority, and it may be accidental that the oldest recorded senses of the noun are attested slightly later than the first attestation of the verb. No etymon suggests itself, and although there is an obvious resemblance of form and meaning in the earlier word nock n.1, which has the parallel senses ‘notched tip of a bow’ and ‘cleft in the buttocks’ (compare sense 2b), it is not easy to see how they might be related.
The OED asserts that nick and niche are "almost certainly not related".
Solution 3:
The relation of "nick" to naked seems relevant (and untouched upon in this conversation). Such a sense could apply here as "essence" or "core truth"; and thus correlate and conflate with "nick" as "notch": as "in the notch" or "in the groove" or "in the slot".
Shakespeare, around 1590, has Nick Bottom the Weaver as a comic character. "Bottom" was a term for what would with industrialization become the weaver's bobbin- a nicked or notched stick of wood around which yarn was wound. Bottom also (or thus) served metaphorically as "stamina" (an available reserve); this sense, though now becoming archaic, was used especially to describe (like "nick") horses and their abilities and congress; and appropriately Nick Bottom was the man who, in Midsummer's Night Dream, was turned into an ass (donkey); and who, thus transformed, became a queen's (temporary) lover.
Along with some possible implications of the word "nick", the relationship between "bottom" and "ass" (anatomical) has been explored at length here The Name "Bottom" in A Midsummer Night's Dream
It may also be that "Nick" serves in this instance as a synecdochical form of "neck"; the specific "nick-neck" conveying a sense of carriage in general; certainly how a horse holds its head is an essential indication of its strength speed and general conduct.
HEAD & NECK The head and neck are important in determining the athletic ability of the horse. A supple horse uses its head and neck as a rudder and stabilizer. Free head and neck movement has a profound influence on the horse’s way of going. from Horse Conformation Analysis