Help! Trying to find an Anime for a Person who doesn't watch anime!

Some of these have been posted before, but meh, this will be my list regardless :)

Cowboy Bebop. I know you said it is dated, it is a little bit but not much. Has decent pacing, amazing English dub, and doesn't have the fanservice and the like that often turns people off. Also the Movie is amazing even as a standalone.

Black Lagoon. Can't really go wrong here.
Ghost in the Shell. Depending on his tastes, either the movie, SAC, or Arise. All are good.
Planetes. If they want something with more story than action.
Durarara.One of my favs.
Psycho-Pass.
Appleseed Movies are another good starting spot.

I used to say Hellsing but both the series and Hellsing Ultimate go all insane at the end imho.. though the first few eps, especially of Ultimate, are amazing.
Don't agree with Deathnote, but then again I didn't like it myself so can't really recommend it :p

SAO is another easy one for people to get into imho, overhyped by some but still a really good anime.

And thats the top of my head.

Check out these recommendation charts by lukeatlook. Some of the ones in this thread are included but it also has other anime that might not have been.

https://imgur.com/gallery/nnG8W
https://imgur.com/gallery/wfiy4eX
https://imgur.com/gallery/ANQyBqn

The Future Diary should be great for them.

one punch man!

Yugioh?
Shaman King?
Sex Taxi?