It depends on what the person likes. If I were trying to convert someone who would like romantic comedies, I'd choose Coffee Prince.
If I were trying to convert someone who'd like history, I'd choose Tree with Deep Roots.
If I were trying to convert a person who'd like a comedy that isn't just a romcom but has other elements as well, then I'd recommend History of a Salaryman. I also think this one would be not so weird in the mind of a person used to American dramas.
If I were trying to convert a person who likes something with fantasy elements I'd do either Arang & the Magistrate or Queen In Hyun's Man (and that would be my choice for someone who likes pure romance too).
Finally, if the person liked suspense AND didn't want to watch something too long, I couldn't recommend White Christmas enough. It's only 8 episodes. Also, if they are used to American tv and film then I think White Christmas would be perfect. It has really, really high quality directing. The soundtrack too would be familiar (with a lot of English songs) and the acting is really good. None of the characters are the goofy, over-acting types that you often see in Korean (or any Asian) dramas as stock clown-types. It is the most American feeling drama I think I've seen out of over 100 Korean dramas.
Honorable mentions, depending on the person's interests: Vampire Prosecutor (suspense, vampires, mystery), Reply 1997 (comedy, family, school) and Painter of the Wind (historical, art, aesthetic).
I hope this helps because I too am really interested in getting people converted to Asian dramas (well, mostly Korean because that is what I watch).