The Rule of Two was developed as a way for the Sith Order to
Survive
Weed out the weak
The whole mechanic of the Apprentice killing the Master allows for only the strongest to survive and advance the order. That reminds me of Natural Selection and made me wonder if Lucas based it on that.
No, it wasn't really about strengthening the Sith through natural selection. The basic rationale for the rule of two was that the Sith tend to struggle for power with each other, which had led to a disaster where nearly all had been killed, so with only two they could spend most of their time plotting against the Jedi instead of infighting. A master/apprentice relationship is more stable because the apprentice will for many years avoid trying to kill the master since the apprentice still has more to learn, and the master wants to have an apprentice to continue the Sith plot against the Jedi if the master should die for some other reason, such as old age (the master also presumably enjoys having a powerful flunky to do their bidding, and is confident enough in their greater abilities to keep them safe if the apprentice should ever rebel).
The first in-universe explanation of the origin and rationale for the rule of two was given in Terry Brooks' novelization of The Phantom Menace, on pages 138-139 which are available on google books here:
The rogue Jedi who had founded the Sith order was its nominal leader, but his ambition excluded any sharing of power. His disciples began to conspire against him and each other almost from the beginning, so that the war they instigated was as much with each other as with the Jedi.
In the end, the Sith destroyed themselves. They destroyed their leader first, then each other. What few survived the initial bloodbath were quickly dispatched by watchful Jedi. In a matter of only weeks, all of them died.
All but one.
...
It was patience that had saved the Sith order in the end. It was patience that would give them their victory now over the Jedi.
The Sith who had survived when all his fellows had died had understood that. He had adopted patience as a virtue when others had forsaken it. He had adopted cunning, stealth, and subterfuge as the foundation of his way—old Jedi virtues the others had disdained. He stood aside while the Sith tore at each other like kriks and were destroyed. When the carnage was complete, he went into hiding, biding his time, waiting for his chance.
When it was believed all of the Sith were destroyed, he emerged from his concealment. At first he worked alone, but he was growing old and he was the last of his kind. Eventually, he went out in search of an apprentice. Finding one, he trained him to be a Master in his turn, then to find his own apprentice, and so on to carry on their work. But there would only be two at any one time. There would be no repetition of the mistakes of the old order, no struggle between Siths warring for power within the cult. Their common enemy was the Jedi, not each other. It was for their war with the Jedi they must save themselves.
The Sith who reinvented the order called himself Darth Bane.
Also, page 4 of this interview with George Lucas from 1999 (the year The Phantom Menace came out) includes a comment where he explains the reason only two Sith can ever stably exist, again because of their tendency towards infighting and struggle for power:
One of the themes throughout the films is that the Sith lords, when they started out thousands of years ago, embraced the dark side. They were greedy and self-centered and they all wanted to take over, so they killed each other. Eventually, there was only one left, and that one took on an apprentice. And for thousands of years, the master would teach the apprentice, the master would die, the apprentice would then teach another apprentice, become the master, and so on. But there could never be any more than two of them, because if there were, they would try to get rid of the leader, which is exactly what Vader was trying to do, and that's exactly what the Emperor was trying to do. The Emperor was trying to get rid of Vader, and Vader was trying to get rid of the Emperor. And that is the antithesis of a symbiotic relationship, in which if you do that, you become cancer, and you eventually kill the host, and everything dies.
The Rule of Two served two functions:
The first and primary function is articulated well on Darth Bane's (archived) page from starwars.com:
An ancient and legendary Sith Lord, it was Darth Bane who saw that the Sith traditions of old were ultimately a dead end. All too often, squabbling Sith in their bid for power upended carefully laid plans and planted the seeds of their own defeat. After the Sith were decimated by the Jedi Knights of a thousand years ago, Bane enacted the Sith rule of two: there would be only two active Sith at one time -- a Dark Lord to embody the power, and an apprentice to crave it.
The second function is obvious from the nature of the Rule of Two, but is best articulated in Legends:
Under [Bane's] leadership the Sith had been reborn. Now they numbered only two—one Master and one apprentice; one to embody the power of the dark side, the other to crave it. Thus would the Sith line always flow from the strongest, the one most worthy. Bane’s Rule of Two ensured that the power of both Master and apprentice would grow from generation to generation until the Sith were finally able to exterminate the Jedi and usher in a new galactic age.
Dynasty of Evil, p. 9 (Legends novel)
It's true that the second function sounds vaguely like natural selection. For example, Encyclopaedia Britannica says that
In natural selection, those variations in the genotype that increase an organism’s chances of survival and procreation are preserved and multiplied from generation to generation at the expense of less advantageous ones.
However, the second function follows from the first: the Rule of Two was instituted to stop infighting, and the practice of the apprentice killing the Master was mostly a consequence of the drastic reduction in the numbers of the Sith. The first/primary function doesn't have anything to do with natural selection, so it doesn't look like the Rule of Two was based on natural selection.
Furthermore, the Sith Order's behavior prior to the Rule of Two arguably looked like natural selection as well -- some Sith were "selected" by the Force to be stronger, and these stronger Sith were able to kill off the weaker ones. And there's no reason why a Sith Order composed of many Sith Lords couldn't institute a rule that each Sith apprentice must kill his Master so as to strengthen each Sith from "generation to generation".
Lastly, I can find no mention of natural selection in the context of the Rule of Two in any Star Wars source (various novels about the Sith, Wookieepedia, starwars.com, etc.).
I don't know about Lucas's idea himself, but it's very probable that it's based on natural selection, although it could also be so there's no power vacuum when the apprentice kills the master, if there were more there would be more chaos, the sith want control, so it makes sense that it would be only two
