This could've been a 'running commentary' department presents:
In English we name the days of the week after a Germanic god – The Sun, The Moon, Tiu, Woden, Thor, Freya – except Saturday, which is named after Saturn, who is not German but Roman. Why is this?
Since ancient times, the skies were known to have the stars (which did not move), the Sun (the big glowy thing), the Moon (the thing with phases), and the five planets Mercury, Venus, Mars, Jupiter, and Saturn (which wandered around the sky on their own agenda).
Why do the planets share their names with Roman gods? Many of the explanations online suck. Take the Washington Post, who in 2016 actually told their readers "Jupiter [the planet] shares a title with the king of the gods because it's the solar system's giant" even though the ancient civilizations that named the planets had absolutely zero concept of the true nature of planets, let alone which of them would be shown millenia later to be the biggest.
All that ancient civilizations knew about the planets was how fast they appeared to move through our skies. Mercury, which we now understand to be a small rocky planet near the sun with a correspondingly quick orbit, moves through our sky the fastest among the planets. Hence, the ancient peoples named it after their fastest, the god of messengers. Venus, which shines brilliantly in the dusk and dawn hours, is a good match for the god of beauty. Mars, reddish in hue and prominent in the sky, is again a natural pair for the important god of war. The planets Jupiter and Saturn, which to an ancient observer on Earth are less bright, less quick in their movement, less carrying distinct hue, probably were associated with their assigned gods not out of resemblence, but instead because of the importance of those gods in the pantheon.
It's worth noting that planets are only named after gods in western languages. For instance, Chinese language uses classical elements for planet names.
But in Western tradition, a simplistic answer to the linking of planets to gods is that the Romans named their planets adopting the earlier Greek practice [note: check out this link for detailed, sourced etymology]. And it was Greek practice because they adpoted the earlier Babylonian practice. And it was Babylonian practice because they got the idea from someone before them. We can't know who, but someone, somewhere first drew the line between planets and gods, and it stuck. Why?
I am not a scholar of ancient religions, but I know how to type things into search engines. And I've been searching this question for a long time. The most compelling answer I've ever found is, believe it or not, on Quora, a crowd-sourced answer website of questionable integrity, from someone who assumes no air of authority whatsoever when calling himself an "amateur armchair bloviator". Our bloviator, James Card, says
Like modern people, it's hard to understand or explain what ancient people really thought about their gods. In some sense, the ancients really believed that the planet Jupiter was literally the god Jupiter; and yet simultaneously they didn't believe that at all.
As my favorite history blog reminds the reader ad naseum, "it is generally safe to assume that people in the past believed their own religion" [recommended reading: Practical Polytheism].
So, it happened and we must accept that it happened. Some ancient priest stayed up past his bed time, pointed to the sky, and said, "That bright moving dot is the king of the gods!" and everyone listening to him went along with it. By the time the practice of planets = gods made it to Rome, it was already ancient, and when time came for naming the planets with Roman names, they used the religious syncretism of interpretatio graeca to wholesale import Greek culture, including the planets slash gods, but re-identifing them with the similar Roman pantheon. So the planet Zeus became Jupiter (etymology: dyeu-pater, deus-father), Cronus became Saturn, Aphrodite became Venus, Hermes became Mercury, and Ares became Mars.
Despite that Zeus and Jupiter are not the same god, they did represent similar things to their respective cultures. Both were for instance the king of the gods and associated with thunder. Not all gods translated from Greek to Roman culture so neatly, but these planet gods did, so the planets took the names by which we still know them in English, two thousand years later.
We're getting there.
The seven-day week originated from the Lunar cycle. There is a full moon roughly every 28 days, and that divides roughly into four 7-day cycles, each associated with a phase of the moon (full, waning, new, waxing). This is convenient for ancient nomadic civilizations that needed to schedule meetings prior to the invention of cell phones. Jewish festival days, which frequently begin on full or new moons, are still extant examples of this practice.
When the seven-day week was adopted by Rome, those days needed names, and for that we turned to astrology. In astrology, for reasons only astrologers understand, each hour of the day belongs to a planet. With seven planets and twenty-four hours, the first hour of each day cycles naturally through each of the planets, and it was to that planet that each day was said to 'belong.' For instance, if the first hour of the day was The Sun's hour, then the day was The Sun's Day, or Sunday.
Modern English calls the days Sunday (The Sun's Day), Monday (The Moon's day), Tuesday (Tiu's day, or maybe Tyr's day, or Tiw's day), Wednesday (Wodan's day), Thursday (Thor's day), Friday (Freya's day), and Saturday (Saturn's day). Who the heck are Tiu, Wodan, Thor, and Freya? Isn't Thor that beefy Australian with the hammer?
Just as the Romans interpreted Greek deities into their own culture (e.g. interpreting Zeus as Jupiter) through interpretatio graeca, there was a similar interpretatio romana (or germanica) tying the gods the Germanic tribes already worshipped into the conquering Roman gods. However, unlike the import of Greek gods, which much more closely aligned with Roman pantheon, the Germanic gods were more different. Modern fascination with (Marvel movies or) the Norse pantheon (which was a very late form of Germanic pantheon) teaches us that Odin was the king of gods, Friga was his wife who could see the future, and Thor was Odin's badass son who crushed mountains with his hammer while cracking wise. These Norse gods are better preserved than their Germanic predecessors (who are barely preserved at all), but the pattern of worship is clearly, distinctly different than that of the Greeks and Romans.
That different pantheon didn't stop the interpretatio romana. The Romans made their new Germanic subjects pay their taxes and didn't care who they worshipped. The institution of the Roman calendar week was necessary in order to conduct business, but the names of the days wasn't important so long as they paid their taxes. Hence, using local deities in place of Roman to name the days of the week.
The Roman tax collectors weren't students of their subjects' religion, so the swapping out of deities didn't follow any systematic religious planning, how a modern person might categorize these ancient gods. Rather than matching gods with a similar role (e.g. King, daughter, etc.) in the pantheon, popular gods were selected from what looks to modern eyes to be superficial aspects of their godhood. So Jupiter's day didn't become Odin's day, but rather Thor's day, because both Jupiter and Thor are associated with thunder, and Odin is not.
Mercury's day became Wodan's day, because even though Wodan was the king of the gods and Mercury was not, many of their other aspects are shared, such as being travelers, poets, and having involvement with the dead. Venus's day became Freya's day: Freya is more warlike and "important" than Venus, but they are both women and symbols of love and sensuality. Tyr/Tiu/Tiw is less well known to us despite his importance to the people who worshiped him, but he was a warrior and became associated with Mars. The Sun and Moon connections are straightforward – Sun and Sol, Moon and Luna – being close analogies.
Which leaves just Saturn. Why did Saturn not get interpreted to a Germanic deity when the days were named? Wikipedia claims, without citing a source, that "none of the Germanic gods were considered to be counterparts of the Roman god Saturn". This doesn't make sense, as the other gods were not necessarily close counterparts, and it's not like there weren't other Germanic gods available.
The second-most voted answer on this Stackexchange English thread about this same question, written by someone called Kyle Pearson, claims that Saturn did get associated with a Germanic god, but it was Thor. And then he explains with a whole cascade of other changed interpretatio's and convoluted malarkey that Saturday's origin is in nobody wanting a day of the week named after the trickster god Loki. The thought is that the Loki-Mercury day was dropped and the rest of the days were rearranged, then because they were short a day the people just brought over Saturday from the Romans a second time, this time leaving the name unchanged. This is stupid, and clearly wrong.
To back up this crazy theory, they cite the book "Hamlet's Mill" by Giorgio de Santillana, an MIT professor, and Hertha von Dechend, a "scientist" about who I can find no further information. That's great, except that etymologist Jaan Puhvel said things about Hamlet's Mill like "it obviously solicits the suspension of disbelief", and anthropologist Edmund Leach said things like "I do not believe a word of it". These are just two among the book's many harsh criticisms. So, while possible, my armchair analysis decrees this convoluted, cockamamie theory to be "bullshit".
Scroll a little farther down that same Stackexchange thread to the 3rd most voted answer, authored by "ghoppe", and we have what feels to be the strongest answer to this puzzle. The meat of the post is this:
There's an old Anglo-Saxon poem Solomon and Saturn which is basically a "riddle contest" between the wisest king of the land of Israel and the Roman god, done in the style of Norse poetic eddas. I haven't been able to pin down when it was written, I think in the 9th or 10th centuries. So I take this as some evidence that Saturn wasn't lost to Anglo-Saxon folklore. In the poem, he represented the pagan and eastern tradition, held in contrast to the Christian tradition and faith represented by Solomon. So perhaps this is a case where the name of saturday didn't really change because Saturn wasn't an obscure foreign god, but rather a well-known (although still foreign) entity from folklore.
You can read the translated poem online, but you don't need to in order to understand ghoppe's answer: Saturday wasn't renamed because the Germanic tribes knew who Saturn was.
I'm not a scholar (nor do I own a time machine) so my opinion on this question weighs little, but this answer "feels right". I've lived long enough to know that every answer that "feels right" isn't necessarily so, but in this case I can live with believing that this answer is likely the best I'll get.
Saturday is (likely) the sole day with a Roman name origin (instead of Germanic) because Saturn was a god known to the Germanic tribes despite his foreign origin.
A good part of my fascination with this question is how difficult it was to find a straightforward answer to what seems like what would be a common question. Multiple times a day I look up answers to questions that pop into my brain from nowhere. But with this Saturday question, instead of easy answers like most of my questions have, result after result gives superficial, garbage answers to how the day names came to be. Science websites zip through the etymology of planet names as quickly as possible. Etymology websites focus more on the construction of the sounds than the anthropological "why". And pop-sci websites just say "they're Roman lol". It's weird for me that someone might read such a partial answer to this question and be totally content with that explanation. None of the naming of these or any origins of words are accidents – things have names for reasons, and days are no exception.