Sources conflict:
Brewer's Dictionary of Phrase and Fable says that "the practice was popularized by Russian officers at the Tsar's court in the months before and during the 1917 Revolution, when the term is first recorded." But they don't give a source, and don't say where it was first recorded.
According to an essay in the Comprehensive Textbook of Suicidology, the first known description of Russian Roulette occurred in the writings of Lord Byron, whose roommate apparently played and survived a variant of the game (though it didn't involve a revolver, and wasn't called "Russian Roulette") sometime prior to 1821. According to this source, Byron's account inspired a scene in Mikhail Lermontov's short story "The Fatalist" (1839), which "served as the prototype of for the subsequent practice of Russian Army officers using a pistol in a gamble with death."
Byron's (brief) description of how his friend Edward Noel Long put a gun to his head and pulled the trigger (apparently without knowing for sure whether or not it was loaded) is here:
He told me that, the night before, he had "taken up a pistol - not knowing or examining whether it was loaded or no - and had snapped it at his head, leaving it to chance whether it might, or might not, be charged."
It's too long to quote, but if you're interested, you can read Lermontov's short story here (it's part of a larger work - just do a page search for "the fatalist").
I searched for medical literature on this, most public health scholars suggest that the game originated in tsarist Russia, often citing French sociologist Jean Baechler, who simply states in his 1979 text Suicides that the game was invented by Tsarist officers in Russia. Neither Beachler nor any of the other medical literature I could find referred to a specific historical source for this information though.
This book, agrees that Byron described a similar game, and that this was similar to what's described in The Fatalist, but also says that it's "completely wrong" to see this as the origin of true "Russian Roulette," because in both of these works the gun used was a single-shot pistol, and there's no proof that Lermontov was inspired by Byron.
According to the Oxford English Dictionary, the earliest known use of the term "Russian Roulette" is actually from 1937, when a short story by Georges Surdez titled "Russian Roulette" was published in Colliers Magazine. The relevant passage is:
Did you ever hear of Russian Roulette?... With the Russian army in Rumania, around 1917,... some officer would suddenly pull out his revolver,... remove a cartridge form the cylinder, spin the cylinder, snap it back in place, put it to his head and pull the trigger." (Quote & ellipses taken from OED, not the original source)
A quick search of Google ngrams pretty clearly suggests that the term "Russian Roulette" was not really used at all prior to the 1940s (after Surdez's story was published).
Graham Greene claimed in his autobiography (1971) to have played the game himself in 1923. He claims that he found a revolver and:
I knew what to do with it because I had been reading a book (I think Ossendowski was the author) which described how White Russian officers, condemned to inaction in southern Russia at the tail-end of the counter-revolutionary war, used to invent hazards with which to escape boredom. One man would slip a charge into a revolver and turn the chambers at random, and his companion would put the revolver to his head and pull the trigger."
Graham is referring there to Ferdynand Antoni Ossendowski - who did indeed serve in the White Russian forces and later write about it. The kicker, though, is that nothing like what Greene describes actually seems to have appeared in any of Ossendowski's writings (at least, not as far as I - or any source that I can find - has established). So this may be a case of Greene mis-remembering and/or projecting something backward (a pretty common occurrence in memoirs).
To sum up: there are lots of hints of a Russian origin, but no direct evidence. To me, and taken as a whole, these sources suggest that the term Russian Roulette originated around the time that Surdez published their story, and was popularized in the years that followed (which is when I also found the first/earliest newspaper reports of people being killed playing "Russian Roulette"). Notably, newspaper databases yield nothing for "Russian Roulette" prior to 1938, when I found several obituaries for people who had shot themselves while playing the game. I don't think it's a coincidence that those kinds of stories appear one year after the publication of Surdez's story.
It's possible that the game itself is older than that (Greene does claim to have played it himself in 1923) but it's difficult to say for sure from any of the sources I have looked at or can find. Since it appears in the Colliers short story itself, we should at least consider the possibility that the "tsarist officer" angle may very well have been an invention of Surdez too, and that Greene was projecting that backwards. But it's not possible to say for sure - especially since if the game existed before 1937, it appears to have been known by a different name (making it hard to find sources that document it).
tl/dr: It's often claimed/written that "Russian Roulette" originated with Russian officers around the time of the Russian Revolution, but there's no direct evidence that confirms this that I can find. It's also often claimed/written that either Lord Byron or Mikhail Lermontov first wrote about the game during the first half of the 19th Century - but neither of them actually referred to "Russian Roulette," and both described a game involving single-shot pistols. There is considerably more, and more direct, evidence that a 1937 story by Georges Surdez published in Collier's magazine was what popularized the idea of "Russian Roulette" as we know it - the term itself does not appear to predate that story, and newspaper accounts of people actually playing the game only start to appear the following year (when they're surprisingly, and tragically, common for several decades afterwards).
Hi, more discussion is welcome here, but do check out this earlier thread
To complicate matters further, when I was reading Moshe Gammer's The Lone Wolf and the Bear (Pittsburgh, PA: University of Pittsburgh Press, 2006) about the Russian conquest of Chechnya and the North Caucasus, I came across this footnote:
...Caucasian Mountaineers were not conscripted into the Russian army, although volunteers were welcome. In addition to the above-mentioned two Dagestani and one Chechen irregular mounted regiments, the Emperor’s guard included a Mountaineers unit, and a regular army division was formed known as “the Wild Division” (dikaia diviziia), known among other things for the invention of “Russian roulette.” (108)
Gammer was considered the foremost authority on the North Caucasus in the English language (he died a few years ago unfortunately), and his book was meticulously sourced, so I don't doubt the validity of his statement. However, other than a casual mention in a footnote, he never brings Russian Roulette up again, nor goes into detail about the Wild Division. Perhaps it comes up in one of his other books, but this is the only one I've read by him thus far.
Timeline-wise it does fit the other explanations of the Tsarist Russian period. Gammer's referring to the time after the Chechens (and other Caucasian peoples) had been officially conquered and were living uneasily beside the Russians. This puts the period in the late 19th-early 20th centuries, before the Bolshevik Revolution. However, I'm not confident the Caucasians themselves invented Russian Roulette, as there was (and still is) a longstanding Russian tradition of attributing excessive barbarity to the mountain natives when they weren't especially more barbaric than the Russians themselves – and often were simply reacting to the brutal slaughter and subjugation by the Russian troops. So it's possible the practice was simply attributed to them for this reason. Without more information, it's difficult to know, but I wanted to throw in the tidbit I had gleaned in my own research.
On a final note, as mentioned in a previous comment, Mikhail Lermontov did write about the Caucasian conquest, so he might have had some insider knowledge he was pulling from. Also quoted in Gammer's book:
The scenes of looting and carnage that followed the conquest of native auls [villages] were described with great disgust by Tolstoy and even in stronger terms by Lermontov:
Auls are burning, their defenders mastered,
The homeland’s sons have fallen in battle.
Like steady comets, fearful to the eyes,
A glow is playing ‘cross the skies,
A beast of prey with bayonet, the victor
Charges into the quiet dwelling.
He kills the children and the old folks,
And with his bloody hands he strokes
The unmarried girls and young mothers.
But woman’s heart can match her brother’s!
After those kisses, a dagger’s drawn,
A Russian cowers, gasps – he’s gone!
“Avenge me, comrade!” And in just a breath
(A fine revenge for a murderer’s death)
The little house, delighting their gaze,
Now burns: Circassian freedom set ablaze!
Mikhail Iur’evich Lermontov, “Izmail Bey,” Pt III. Translation by S. Layton, Russian Literature and Empire: Conquest of the Caucasus from Pushkin to Tolstoy [Cambridge University Press, 1994] (52-53)
It's not Russian Roulette per se, but Douglas Porch in The French Foreign Legion notes that Russian recruits into the legion just after the First World War, "refugees from the White Russian armies defeated in the civil war" were credited with the creation of a game in the interwar Legion called coucou that seems to be very similar to the Tiger/Hunter game talked about by u/TheYottleInTheBottle:
"Apparently imported from Czarist army messes, coucou was a variation upon Russian roulette with the odds of survival greatly reduced. The rules required one legionnaire chosen by lot to step outside while his comrades arranged themselves around the walls of a completely darkened room. The single legionnaire then reentered the room and said 'coucou', while, one imagines, simultaneously taking violent evasive action as his comrades took this as a cue to fire their revolvers in his general direction." He cites this game from the testimony of a Legionnaire called Weygand.
Sources: Porch, Douglas. The French Foreign Legion. New York: HarperCollins Publishers, 1991
Weygand, Jacques. Légionnaire. Paris: Flammarion, 1951