First Doctor: The Aztecs, probably. It's not even my favorite Hartnell historical (that would be The Reign of Terror but until that one comes out with animated bits it's sort of dreadfully a bad idea, and it's a bit longer than it needs to be. The Aztecs is only a 4-parter, has Ian and Barbera being awesome, and takes a pretty traditional stance on Time Travel for complete newbies to the show. Also, for New Who fans who are into the more fluffy stuff, it has adorable Doctor/Cameca romance.
Second Doctor: This one is harder -- I like Troughton, but I am not as ALL ABOUT THE TROUT as a lot of classic Who fans, and so much of his stuff is gone. If I'm reccing to a New Who fan who likes Cybermen, The Invasion is a clear choice. But it's a bit long -- you know, still going to go with it. Iconic monsters, lovely supporting cast, must be the remastered animated version. If someone is daunted by eight episodes, then The Ice Warriors, with the nice synopsis of Eps 2&3 that I think is from the VHS edition. It also has Jamie, it has the lolariousness of modern industry leading to Global Cooling menacing the earth, and fabulous set design in places.
Third Doctor: Everyone else I've seen do this meme has picked The Mind of Evil and it's my choice as well. The Master being especially good, Jo getting one of her best outings, and Pertwee does a good job as well. If I'm reccing to a New Who fan that doesn't like the Master and/or for some unknown reason liked the New Who Silurian episode, The Silurians isn't a bad choice, but it gets kind of long and very serious at times.
Fourth Doctor: For all that Baker is The Iconic Doctor for a lot of people, picking one ep is kind of tricky. Though if reccing to someone who likes Douglas Adams, The Pirate Planet is worth doing a quick Key to Time primer first. If they're more into serious stories, want Daleks or Sarah Jane Smith due to New Who exposure -- Genesis of the Daleks. But it's kind of heavy for a complete newcomer. The Sontaran Experiment might be good for a New Who fan who's not certain about really long serials.
Fifth Doctor: This one's easy. Enlightenment, even if it does require a little information about Turlough and the Black Guardian. It's a fun idea, silly but not too silly, the main cast all turn in lovely performances, and the special effects mostly hold up. I haven't seen the edited-streamlined version from the DVD -- if it's well done, it might be a good choice.
Sixth Doctor: Hmm. I like Colin Baker much better in the audios -- he is given such a limited range to work with on the screen. If it were possible to have a de-Trialed Terror of the Vervoids I'd pick that -- Mel and Colin actually have a lovely chemistry in it, but explaining the trial = a no go. So probably Mark of the Rani, which stands out amongst his episodes for being funny. The rest are just bleak, though for the right audience Vengeance on Varos might go over well.
Seventh Doctor: Survival! Short, sweet, has all of the elements of team Seven and Ace plus the Master in one of his better outings. I love Curse of Fenric but I think it's better in context -- the whole thing with Ace's mom and stuff.
Eighth Doctor: Not much choice here, unless I'm diving into the audios or books, which I'm not.
Ninth Doctor: I still think Rose has held up pretty well, even if not everyone does. The effects are awful, but it was designed to be an intro, and does a good job of it.
Tenth Doctor: If someone wants to get into New Who, I think there's little enough to point them at Eccleston and not Tennant, but if I had to pick one, I think I'd either pick Partners in Crime or Smith and Jones Actually, The Runaway Bride might not be a bad choice either, if Christmas Special-ness isn't a turn-off. All of those were intended to do character introductions (or re-intros) and don't get too timey-wimey. (I like timey-wimey stuff, in reasonable doses, but I'm assuming a potential viewer who may not be a big SF fan.)
Eleventh Doctor: Again with the intros -- The Eleventh Hour is really the best choice.