I will look specifically at the western Classical origins of these terms. The Ancient Greeks in particular had extensive mythology and naming associated with all the Winds (Ἄνεμοι Anemoi) and directions, not just the cardinals. The Greeks were a seafaring people and wind direction was central to their lives.
North
L septentriō (adj. septentriōnalis) = septem "seven" + triō, that is "the seven plough-oxen (stars of Ursa Major). triō (pl. triōnes) is problematic. Most sources give this as meaning "plough-oxen", but this term for plough-oxen is used nowhere else in Latin. A few sources posit that triō < PIE *(s)tē̆r- "star" with loss of initial s as in Indic, and that later mythology led people to reinterpret the root as "oxen". The transparent meaning of this word to the Romans was "in the direction of the constellation the Plough (Ursa Major)".
L boreās (adj. boreālis) was also used to mean "north" or "North Wind" and was a direct borrowing from Greek Βορέας. The Romans also called this Wind Aquilō; this latter word is of unsure etymology. There have been attempts to relate it to aquila "eagle", aquilus "dark" and aqua "water", viz. "rainy wind".
Gk Βορέας boreas "north, the North Wind" was a Greek word also of unsure etymology. It has cognates in other Balkan and Slavic languages such as Alb borë "snow", Srb бура "cold north wind". It is often said to come < PIE *gʷor- "mountain". This very likely is a reference to a North wind, cold and perhaps arising from mountains, that was prominent in these people's original homeland.
South
L merīdiēs (adj. merīdiōnālis) meant "noon, midday" < medius “middle” + diēs “day”. Since the sun is in the South at midday in the Northern hemisphere, this word is self-explanatory.
L auster (adj. austrālis) was the Latin name of the South Wind and the South. Now here is a fascinating bit of history and its relation to language change. Most scholars believe that auster < PIE *-aus "shine" - which is the same root that gives rise to the words for "dawn" and "east" in other IE languages! How can the same root be used for different cardinal directions in sister languages? One theory is that since the Italian peninsula runs diagonally NW-SE, the word for "east" shifted to mean "south" since both were in the direction of the Tyrrhenian Sea. Another theory is that since the lands to the South were burning hot, or alternatively since the sirocco was hot, the root *aus- referred to this heat.
Gk νότος notos was the South wind of the Greeks. I have no idea what its etymology is.
East
L oriēns (adj. orientālis) was the usual Latin term. The meaning was transparent in Latin: "rising", viz. "in the direction of the rising sun". oriēns is the present participle of the deponent orior, "rise" < PIE *or- O-grade of "move"
Gk ἠώς eos was used in Greek and also meant "dawn", which is cognate with Latin aurora and Germanic east. ἠώς < PIE *h₂ewsṓs/*h₂ausōs < *aus- "shine". This is conjectured to be a reference to the shining dawn; but see 'south'.
West
L occidēns (adj. occidentālis) was the usual Latin term. The meaning was similarly transparent in Latin: "going down/setting", viz. "in the direction of the setting sun". occidēns is the present participle of occidō, "fall/go down" < ob “towards/facing” + cadō "fall” < Proto-Indo-European *ḱad- “fall”.
L vesper "evening" was also used to mean "west" in reference to the setting sun.
Gk ἕσπερος hesperos was found in Greek, cognate to Latin vesper and Germanic west. ἕσπερος < PIE *wesperos/wekeros "evening" < *wes "wind, blow" + *pero "source". In origin this may have been something like "the direction from which the wind blows"; one can imagine that this is possibly a reference to the prevailing winds in the PIE urheimat.
