Before we get into the original answer, please note that /u/MiLiLeFa actually found a site documenting even earlier examples – comment here. While it seems like modern 'slime girls' do indeed trace back to the rise of fantasy settings in Japan in the late 1980s as I had been led to believe, a few predate what I thought was the first, and the ur-examples of the trope look to actually be from the 1960s, but with a considerable lull between the late 60s and mid 80s.

So. Uh. I guess I am actually writing this. Let me preface this by saying I have close to zero expertise in late 20th century popular culture... anywhere, let alone Japan. And yet somehow I have wasted spent much of my evening looking into this because... just because. And here's what I've found, generally speaking. There is, as far as I could gather, no real academic literature on the history of this particular type of creature in media, so this will, by its nature, involve a lot of original research based on English-language Internet trawling because, well, nihongo wakarimasen. Be warned that a lot of the content being discussed may be considered very NSFW depending on your standards, especially the post-2000 stuff.

Now obviously there is a 20-year-rule here but I feel like as this is basically apolitical, it's fine if I give context by noting that there has definitely been a substantial boom beginning in the early 2010s in terms of varyingly sexualised series revolving around 'monster girl' characters in manga, anime, and light novels; examples on the the lewder side include Monster Musume (manga 2012-, anime 2015) and Monster Girl Doctor (light novel 2016-, manga 2018-, anime 2020); examples on the more family-friendly side include Interviews with Monster Girls (manga 2014-, anime 2017) and the slice-of-life (and also, at least in its anime series, social commentary) series A Centaur's Life (manga 2011-, anime 2017), although your mileage may vary as to whether the latter ties in more closely with the much longer-standing kemonomimi (animal ears) trope. However, there are examples predating Monster Girl Quest, which released in March 2011, as one of the first works in this boom in 'monster girl' content was the also erotically-themed Monster Girl Encyclopaedia, which was released at Comiket 79 on New Year's Eve 2010. But even then, neither work was the first in Japan to feature the 'slime girl' archetype.

There's definitely a couple of potential precedents from the noughties: in 2007, the character of Melona, with slime-based powers, was featured as one of the two main characters in the eighth entry of the Queen's Blade series of CYOA books, and is also prominent in its two anime series from 2009 and its subsequent OVAs. We get into 20-year-rule compliant territory with the second One Piece film, released in 2001, featuring as its main antagonist Honey Queen, whose Devil Fruit-based ability involves being able to manipulate as well as transform into an unspecified pink liquid. The trouble with these examples, however, is that they don't quite fit the bill of what might be termed the 'modern' 'slime girl' archetype, in which there is a basically humanoid shape but composed of coloured slime. Melona in Queen's Blade generally mimics the appearance of human skin except for having clearly slime-based hair, while Honey Queen in One Piece generally walks around as a normal human; neither adopts a humanoid shape in liquid form, and both can use these slime powers in their humanoid form. This also seems to apply to the character of Isaka Minagata from the mecha manga Cannon God Exaxxion (1997-2004), who is able to shapeshift owing to being made up of nanites, but generally takes a recognisably human form.

That isn't to say that there aren't examples close to the 'modern' slime girl before 2010, and there are a few examples around. However, the two pre-2001 cases that were easiest to locate based off English Internet searching (which led me to a Vice article and a TVTropes page, so, take it with a grain of salt that they are as prominent as they make out) both come from episodes of Sailor Moon. The first, chronologically, was Jamanen, a sort of T-1000-esque robot character composed of a translucent red gel who appeared in the August 1993 episode 'Dispute Over Love: Minako and Makoto’s Conflict', while the December 1995 episode 'Pegasus's Secret: The Boy Who Protects the Dream World' featured PeroPero, a shapeshifter composed of a pale green hard candy. The former in particular seems to carry most of the hallmarks of the 'modern' 'slime girl' archetype given her physical appearance and ability to shapeshift freely, and may be the earliest widely-known example; as of writing I don't know of anything older fitting this particular bill. It is worth considering that the T-1000 comparison may have substance to it: Terminator 2 had come out just two years before the episode featuring Jamanen, so it's definitely plausible that the relatively family-friendly Sailor Moon was riffing on that, but having a gel-based shapeshifting killer robot instead of a metal one.

Sailor Moon's depiction of 'slime girls' may well have influenced a couple of characters later in the '90s. The series Cutie Honey Flash, which aired 39 episodes in 1997-8, featured in its fourth episode a shapeshifting enemy named Bubble Claw, which took the form of a woman with a purple skinned torso, mechanical legs, and clearly slime-like translucent blue-green hair. Cutie Honey Flash was made by many of the same people who had worked on the final arc of Sailor Moon's original five-season run, and so Bubble Claw is quite likely to have been another spin on the shapeshifting female enemies previously featured on Sailor Moon. But aside from this in-house case, we also see, possibly, the first outside derivative of Jamanen in the form of Tiara's demonic doppelgänger in the fifth and sixth episodes of the OVA miniseries Shamanic Princess (1996-8), who also takes the form of a red shapeshifter, though one that does not tend to have a noticeably 'slimy' form when shifted.

If we want to throw a curveball in and stop restricting ourselves to girls as such, there's also the shapeshifting Majin from Dragon Ball, the most prominent of whom is Majin Buu, first featured in the manga in chapter 484 in August 1994, and in the anime in episode 254 in February 1995. Obviously calling Majin Buu a 'slimegirl' might not be entirely apt, but you could potentially shoehorn him into the role – at least, whoever maintained the 'Slime World Wiki' did when they listed him under the section on anime examples of the trope. But in any case, the Majin postdate the appearance of Jamanen on Sailor Moon, so they definitely wouldn't be the first notable shapeshifter characters in anime.

Yet there is at least one example, even older still, of some kind of 'slime girl' character: the somewhat obscure fantasy parody manga Dragon Half, released between 1988 and 1994 and with a two-part OVA adaptation in 1993, introduced the character of Princess Vina in Chapter 3, who would go on to be the protagonist Mink's rival for the affections of checks notes dragon slayer-turned-pop idol Dick Saucer. Vina doesn't quite fit the conventional 'slime girl' bill, as she doesn't really operate as a person visibly made of slime; instead, she transforms between a human form and that of a small spherical slime blob. As far as I have been able to determine, then, Vina is the earliest slime girl, even if only in the semantic sense that she is a slime and she is a girl, but at different times.

This did lead me to then seeing if I could work out where the idea of having slimes at all, let alone 'slime girls', might have come from. A plausible and often-cited contender for the introduction of slime enemies into fantasy media is Dungeons and Dragons, which has featured variously sentient slime-based creatures known as Oozes, perhaps most famously the Gelatinous Cubes, beginning with the original 1st edition in 1974. D&D 1e had been translated into Japanese in 1985 and so may have influenced the various creators of anime slime creatures in the years following. D&D later introduced slime creatures that can take on partly humanoid form with the Ghaunadan, which seem to have featured only in 2e and 3e; however, given that the former was not released in the US until 1987, it's unlikely that our first known example, Vina in Dragon Half, was influenced by the Ghaunadan in particular. Plus, the Ghaunadan are generally depicted forming male-presenting upper bodies.

However, one area we haven't looked at is video games. The Dragon Quest series, beginning with its first entry in 1986, has famously featured small, teardrop-shaped slime blobs as early-game enemies. Given Dragon Half's roots in parodying manga and Japanese fantasy tropes, the depiction of Vina's slime form as a small spherical blob was almost certainly a reference to that version of the humble slime. But having her have a slime form and a human form seems to emerge organically out of the series' own conceits, as she was born from a human father and disguised slime creature, mirroring the protagonist Mink being the child of a human father and dragon mother. Unlike later versions of the 'slime girl' character, the slime form here is basically that seen in Dragon Quest, as it cannot shapeshift in and of itself, only to the human form and back. This is also true during flashback sequences showing Vina's mother, who was originally a slime creature but drank a 'human potion' to assume human form. As such it's not likely that Dragon Half exerted much actual influence on the familiar 'slime girl' archetype we see today, but does serve as an interesting sort of prototypical moment.