What was the alignment of the Elder Things?
That depends largely on which stories you're talking about.
The Elder Things, like most of the other extra-terrestrial creatures in Lovecraft's original stories, can't really be described as "good" or "evil" in the traditional sense. They were alien, so ascribing human motivations to them is somewhat pointless. For the most part, they simply existed on Earth, much like people do now, living their lives.
On the one hand, At The Mountains of Madness claims that the Elder Things were "enemies" of the Great Old Ones. Since the Great Old Ones tended to do things like eat people or drive them insane, we would naturally assume their "enemies" were defending humanity from their evil schemes. But I think that's imposing our ideas of good and evil onto two groups that barely register humanity as worthy of attention. Instead, what we're interpreting as "good" and "evil" is the simply happenstance that the Great Old Ones way of life causes bad things to happen to us and the Elder Things way of life doesn't.
One things that counts strongly against the Elder Things being "good" is their enslavement of the shoggoths. Even after they knew the shoggoths had developed sentience, and desired freedom, they kept them enslaved. On the other hand, this is one of those difficult moral questions that we still haven't got a good answer far, only these days, we ask it about humans and machines instead of Elder Things and shoggoths.
Later, once Derleth took over and started expanding the mythos, he retroactively imposed a more clear-cut distinction. The Elder Gods were a force for "cosmic good" and opposed the Great Old Ones' "cosmic evil", and that was that.
The question is inapplicable. The creatures Lovecraft described were generally operating on their own morality and customs, without reference to humanity. The philosophical basis of much of his work was that humanity couldn't comprehend what was going on in the world. When encountering things humanity was not meant to understand, humanity in general lost, but that's humanity's weakness instead of hostility from others.
For example, Cthulhu is never described as evil, if you will attribute Cthulhu's eating of some seamen as hunger rather than morals. Cthulhu being alive caused a wave of nightmares and insanity, but that wasn't Cthulhu's intention. Human cultists were evil by human standards, but that's part of Cthulhu and the human race being basically incompatible.
Lovecraft didn't divide creatures into Great Old Ones and Elder Gods; that was August Derleth, who made Lovecraft popular, and who didn't understand Lovecraft's thinking (there's a Lovecraft letter somewhere to that effect). I'd strongly recommend reading the work of S. T. Joshi on these topics.
As far as I know, the first use of "alignment" as you're using it was the original Dungeons & Dragons rules, published in 1974, dividing people and creatures into Lawful, Neutral, and Chaotic. This system was taken from Michael Moorcock's fictional universe, much as the magic system owes a lot to Jack Vance. At that time, nobody took D&D seriously in any way, including Gygax and Arneson. It was just a fun game that was not designed to be anything else. The Good-Evil axis was added later. Alignment is not a useful concept for anything outside D&D and things influenced by it.
FWIW, my uncle is a well respected amateur Lovecraft scholar and enthusiast. I passed this question along to him. He was delighted, as he recently presented a paper at the annual Lovecraft Forum, which he describes thusly:
The last paper I presented discussed the distinction between the Elder Gods and the Great Old Ones. Lovecraft was never specific about their attributes or their physical appearances, except in the case of Cthulhu.
Several scholars have speculated that Lovecraft's descriptions of both the Elder Gods and the Great Old Ones anticipated an event in Judao/Christian mythology, when God threw Lucifer and the other rebel angels out of Heaven to reign over Hell.
Others have seen parallels, comparing Great Cthulhu with one of the four major forces of Nature (earth, air, fire, and water). Thus, Yog-Sothoth represents earth, Cthulhu stands for water, Cthugha of the Interstellar Spaces represents air, and a 4th Great Old one (whose name I forget) stands for fire.
One of my colleagues, Ann K. Schwader of Westminster, near Denver, Colorado, whom I regard as the most brilliant poet who ever lived on this planet, reminds me that when Lovecraft discusses geological tectonic mass shifts in At the Mountains of Madness, he is anticipating Science Fiction more than Supernatural Horror. Fans tend to forget what Science Fiction means -- speculatory literature that discusses the literary uses of advancements in science, interstellar travel, contact with sentient species, and dystopian themes describing the end of the world.
So there you have it, make of it what you will. The answer seems to be- depends on your interpretation. As wiggly as that answer seems, remember that the idea was that these creatures were so truly alien that to even try to conceive of them sent one mad because they were so far outside of our everyday reality. If you think that sounds ridiculous, try learning quantum mechanics, and you'll get an idea of how different the world can really be from your everyday reality.
The Elder Things (assuming this refers to the aliens that feature most prominently in At the Mountains of Madness, in which they are referred to as "Old Ones") are by far the most ordinary aliens in H. P. Lovecraft's stories. They are incredibly tough, but they are made entirely of ordinary matter and are ultimately vulnerable to any kind of concerted mundane attack. The do not have have powerful psychic abilities and communicate simply using sound. They do not owe allegiance to any vaster power bent on interplanetary domination.
The Elder Things created life on Earth, but then paid relatively little attention to it, except for their shuggoth minions, their herds if livestock, and any creatures that they deemed strong and vicious enough to be dangerous. Otherwise, their attitude was largely one of benign neglect. They were sometimes conscientious about their organic technology and sometimes careless, just as we humans are with our own technological developments. In fact, the Elder Things are just as fallible or potentially moral as humankind.
They had not been even savages—for what indeed had they done? That awful awakening in the cold of an unknown epoch—perhaps an attack by the furry, frantically barking quadrupeds, and a dazed defence against them and the equally frantic white simians with the queer wrappings and paraphernalia . . . poor Lake, poor Gedney . . . and poor Old Ones! Scientists to the last—what had they done that we would not have done in their place? God, what intelligence and persistence! What a facing of the incredible, just as those carven kinsmen and forbears had faced things only a little less incredible! Radiates, vegetables, monstrosities, star-spawn—whatever they had been, they were men!
The Elder Things are not utterly evil, like Cthulhu and his kin, or even as wicked as the Great Race of Yith. They are not, as a race, good or evil, any more than humans are intrinsically aligned with one or the other.
The concept of alignment doesn't really apply to the Cthulhu Mythos, at least as Lovecraft envisioned it. Think about cats. They're carnivores, but when given the opportunity, they don't just kill their prey quickly and then eat it. They play with it, actively drawing out the prey's death for reasons we don't currently understand. But we don't call them evil for this, aside from the occasional Internet joke: they don't have the capacity for understanding right and wrong as we think of them. Their intellect is simply too limited.
The various cosmic entities of the Cthulhu Mythos are similar, but at the opposite end of the scale. Their intellects (except maybe Azathoth's) are too vast and unlimited: they can't conceive of good and evil as we think of them, not because our scales of morality are such large and grand concepts to them, but because they're such small concepts.
It's the same for the Elder Gods and the Great Old Ones. We don't like being around the Great Old Ones, and this is fair because they eat us, but we don't really know what they want other than to not be hungry. The Elder Gods seem more friendly from our perspective, because they don't consider us to be food, but we don't know why they don't consider us food, or anything else that they want, really. Are they good? Are they evil? Perhaps in their own minds, they might be. Or maybe they don't think in those terms at all. We don't have the information, and even if we did, we probably wouldn't be able to understand it.
Think more simple and purely from the source of Lovecraft's novels and from a point of view of humanity.
Blind idiot god Azatoth is an absolute CHAOS. From one novel all universe is his dream and who cares of his dream? Nobody. So Azatoth is Chaotic Neutral.
Nyarlathotep is the most humanlike of all Lovecraft's pantheon. He tried to cheat Randolph Carter and finally sent him to Azatoth. So from that perspective he is Chaotic Evil.
Other gods, such as Shub-Niggurath, don't care of humanity. But they obviously don't respect people's laws at all. So they are Chaotic Neutral.
Yog-Sothoth from the other side symbolizes "all in one", "the clue", "the gate", " future and past"... - some sort of order which humanity can't comprehend. He (or she) also don't care about humanity, but some cultists can try to use him for their evil plots. Yog-Sothoth' name is also called by cultists in protective spells. But the Universe (Cosmos), which is symbolized by Yog-Sothoth is an anthonim of Chaos, so I think, Yog-Sothoth is Lawful Neutral.