Hmm. Well, he liked Star Wars quite a bit, and while his tastes lean towards action/adventure, he can also appreciate romantic or psychological tales if they're fast-paced and executed well. We have very different senses of humor, so a dramatic episode might be preferable to a lighthearted one. He really doesn't watch a lot of TV, but he's a gamer and enjoys being able to think and interact with ideas.
Not sure if that helps? I think he'll be satisfied with almost any episode; I just want to make sure I don't foil the opportunity if another episode is better. I made him promise to watch one with me after he couldn't pick out Kirk in a photo of the crew. :P If he becomes a Trekkie, all the better, but I'll settle for basic competence!
After reading your post, if I were you, I'd settle for What Are Little Girls Made Of?, Catspaw or That Which Survives.
What Are Little Girls Made Of? is IMO the only great episode of these but the other two has some elements that might appeal to your dude.
Catspaw? As an introductory episode? I would strongly advise against that, it's not at all typical of Star Trek and also, typically ranked near the bottom of ranking lists, I believe.
I don't like it. But I think that it could be perceived as more fast-paced and adventurish, which she says that he likes, and might be one of those episodes who most likely will be able to lure him into the world of Trek.
I can tell you that the first episodes of Trek I saw were those on The Best of TOS DVD (as a kid I only saw TNG and VOY (which I never learned to like) on TV):
1. The City on the Edge of Forever
2. The Trouble with Tribbles
3. Balance of Terror
4. Amok Time