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. Days were named after planets for astrological reasons, and, since ancient times, planets were associated with gods. To read how I got here, continue on.
Before telescopes and astronomy and physics and everything, the skies were known to have stars (which did not move relative to one another, and were clustered into groups), the Sun (which was distinct from a star), the Moon (which was seen as the opposite of the Sun, such as in the idea of the Moon symbolizing nighttime in art, even though anyone can plainly see the Moon also shines sometimes during the day), and five planets (which wandered around the sky on their own agenda).
These planets are famously named after Roman gods, but many of the explanations online for why they are named such are impossible and defy common sense. For instance, the Washington Post in 2016 wrote "Jupiter shares a title with the king of the gods because it's the solar system's giant" even though the ancient civilizations that named Jupiter had absolutely zero concept that the murky reddish dot wandering the sky was actually the largest lump of matter orbiting the sun.
Better explanations work for Mercury (it's movement through the sky was quick, and Mercury was the messenger god) and Venus (the third brightest object in the sky makes sense for a god of beauty) and Mars (the reddish tint bore resemblance to blood, which Romans and Greeks associated with war and violence). Lastly, in my own interpretation of many sources, Jupiter and Saturn were probably designated the names of the other two planets just because of the importance of those gods, rather than any direct symbolism in the characteristic of the planets themselves.
why are planets named after gods in the first place?
To be clear, planets are only named after gods in western languages. For instance, I've read that Chinese 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 [aside: check out this source for detailed, sourced etymology]. But why did the Greeks do that? They similarly adopted the earlier Babylonian practice. But why did the Babylonians do that? All we're doing is passing the buck. Why are the planets named after gods?
I do have a hard time drawing a connection between understanding ancient gods as manifestations of power on Earth and these lights that wandered through the night sky. Like, how can they be both? The Sun and the Moon are clearly their own things in those ancient religions, being deities only as far as every part of nature (e.g. oceans, rivers, wind) was a deity. So why are the Sun and Moon's counterparts the planets tied to gods that already exist elsewhere in the mythology?
I am not a scholar of ancient mysticism, I just know how to use Google. And I've been searching this question for a long time. The best answer I've ever found is, believe it or not, on Quora, a crowd-sourced answer website of questionable integrity, from someone who calls himself an "amateur armchair bloviator", assuming no air of authority whatsoever. 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.
So we'll just have to accept that for some reason we lack the cultural background to grok, 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 else just decided this was fine. By the time this practice made it to the Roman priests, it was already ancient, and when it came time to 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-identified with the similar Roman gods. So the planet Zeus became Jupiter (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 represented mostly the same things to their respective cultures. Both were the king of the gods, and associated with thunder, for instance. Not all gods translated from Greek to Roman culture had the same level of importance or same meaning, but these planet gods mostly did, so there was little change.
what does this have to do with days of the week?
What do planets have to do with days of the week? 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 (mostly) 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.
Those seven-day-cycles need names for the days, and to source those names, the best answer I can determine can be summed up with one word: "astrology". Astrologers thought (and I guess they still think) that the positions of heavenly bodies impact what happens down here in the dirt. In this sense, each day then "belonged" to a celestial item. How the order was achieved requires digging deep into astrology math – so deep I'm afraid for my mental health – so let's suffice to say that the seven days are named after celestial objects for divination reasons.
why does saturday have a roman name?
To quickly recap: ancient gods were personifications of natural forces, those personas had names, those names (or maybe personas) were shared with the planets, the planets gave their names to the days of the week, and the Romans were the last major civilization to name all these things, so we mostly use Roman names for planets even today. Most romance languages continue to use the Roman names for days of the week, but English does not, preferring Germanic names, except for Saturday, which stayed Roman. Follow all that?
In case this article is so long you've forgotten what the names of the days of the week are called, in English we call them Sunday (the day that belongs to the Sun, astrologically speaking), 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). Wait, who the heck are Tiu, Wodan, Thor, and Freya? Isn't Thor that beefy dude with the hammer? What's he have to do with Jupiter?
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 existing Germanic gods into Roman ones. However, unlike the import of Greek gods, which much more closely aligned with Roman gods, 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. These Norse gods are better preserved than their Germanic predecessors (who are barely preserved at all), but the pantheon is clearly, distinctly different than that of the Greeks and Romans.
it's about taxes
That different pantheon didn't stop the interpretatio romana, though. The Romans made their Germanic subjects pay their taxes, but cared little for pushing the non-tax-related parts of the Roman culture. The institution of the Roman calendar week was obviously necessary in order to conduct business, but the Romans didn't care what the locals called the days 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 looking for gods with a similar role in the pantheon, they seemed to swap out gods based on popularity, keying off 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: Freya is more warlike and "important" than Venus, but they are both symbols of love and sensuality. Tyr/Tiu/Tiw is less well documented despite his importance to the people who worshiped him, but was a warrior and became associated with Mars. The Sun and Moon connections are straightforward – Sun and Sol, Moon and Luna – being the same in both cultures.
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". But that's not a real answer, it's just another way of saying "because I say so".
what crazy people say
The second-most voted answer on this Stackexchange English thread, written by someone called Kyle Pearson, claims that Saturn did get associated with a Germanic god, but it was Thor, and then 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. Because that makes sense.
To back up this crazy theory, this Pearson person cites 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 the book like "it obviously solicits the suspension of disbelief", and anthropologist Edmund Leach said things like "I do not believe a word of it", to site two of the more prominent reviews in a slough of harsh criticism. So, while interesting, my armchair analysis decrees this convoluted, cockamamie theory to be "bullshit".
again, why didn't saturday get germanized?
Maybe they just liked it better that way? — They Might Be Giants (admittedly, on a different topic)
Scroll a little farther down that same Stackexchange thread to the 3rd most voted answer, authored by "ghoppe", and I think we may finally have the best solution to this puzzle. The meat of the piece 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 (lord knows I haven't, it's loonnnnngggg), 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. Days were named after planets for astrological reasons, and, since ancient times, planets were associated with gods.
the last paragraph
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.