The Trouble with Tribbles would be my choice--it doesn't take itself too seriously, and you get a good idea of the characters.
Disagree, but personally I think TTWT is overrated. It’s consistently rated among the top five episodes, and I don’t think it deserves that. It’s a good, fun time for Trek fans, but it relies on the viewer knowing and loving the characters. When Spock, stroking a tribble, says, “Fortunately, I am immune... to... its effects...” that’s funny to Star Trek fans, but isn’t likely to make the uninitiated think, “Wow, this is awesome, I’ve got to see more of this series!” Even if he does think that, you’ve given him a rather inaccurate impression of what the series is about. Other than A Piece of the Action, I, Mudd, and The Voyage Home, nothing in TOS has a tone at all like TTWT.
Here are some of the best episodes in roughly chronological order:
Where No Man Has Gone Before
Balance of Terror
The Galileo Seven
Arena
Space Seed
The City on the Edge of Forever
The Devil in the Dark
Errand of Mercy
Amok Time
Mirror, Mirror
Journey to Babel
The Doomsday Machine
The Trouble With Tribbles
A Piece of the Action
The Tholian Web
I’d go with those episodes in that order. By the time you get to the comedies TTWT and APOTA, your friends will be able to appreciate them. If your friends don’t make it that far, those episodes probably wouldn’t have won them over.
If you want the single best episode, in terms of being accessible to those not familiar with the series, being an excellent episode, and being a good barometer of whether the viewer is likely to appreciate the rest of the series — go with The Doomsday Machine.
If your friends are willing to give the series a fair chance, go with the marathon listed above. If they’re not hooked by the time you get through Arena, they’re not going to be fans of the series.