Klein writes that the root is the Shaph'el form of the root כון - "to be, set up, be established." In this post we discussed that earlier root, and we also discussed here a possible connection between שכן and the word sochen סוכן - "steward, agent."
But I realized that there was one additional connection that I did not discuss. Nicholas Oster, in his book Empires of the Word - A Language History of the World (which I've recommended before) quotes the scholar C. F. D. Moule, who writes here that the Greek skēnḗ - "tent" may have been influenced by the Hebrew root שכן meaning "dwelling."
Now that doesn't mean that this is a direct etymology. The Online Etymology Dictionary provides a different derivation:
related to skia "shadow, shade," via notion of "something that gives shade"
But that doesn't contradict Moule's theory. He writes of "Greek words whose use, or at least frequency, may have been suggested by a certain (perhaps fortuitous) similarity of sound or spelling to certain Semitic words." That certainly could be the case here, and we've discussed many times how this has worked in the other direction - some modern Hebrew words were adopted because of the similarity of sound to foreign words (even if they have ancient Hebrew roots - take maksim מקסים meaning "great" and influenced by "maximum" as just one example.)
As the same OED entry quoted above mentions, the Greek skene gave us the English word "scene":
"subdivision of an act of a play," also "stage-setting," from Middle French scène (14c.), from Latin scaena, scena "scene, stage of a theater," from Greek skene "wooden stage for actors," also "that which is represented on stage," originally "tent or booth"
Scene has expanded its meaning beyond just the acts of play, and can now mean "a place or representation of an incident" or "a specified area of activity or interest." Those meanings aren't actually so far away from our understanding of shechina...