Okay.
First of all, Dragon Quest. Everything under the Dragon Quest label with a half decent rating is worth considering (minus the mmo and maybe Rocket Slime since that's kinda out there.) Dragon Warrior games haven't aged well, but the DS remakes should be just fine. I say stick to these at first because its a title with name recognition, something new players can identify by and think of them based off of past experiences from other games after the first. Its just easier for the player.
There was recently a HD rerelease of Final Fantasy 10 and 10-2. Look into that.
If your friend likes any anime, then see if it has a game. This can be hit or miss as quality differs based on the maker. For instance,
http://atlus.com/atlus-titles/attack-on-...ab-panel-1 Attack on Titan, though its an action game.
Bravely Default, this is a great game that takes everything amazing about Dragon Quest 9 and classic Final Fantasy and turns it up to 11. I wouldn't recommend this immediately because the story involves time travel, and while that's not a bad thing by any means, it leads to a lot of back tracking that can turn off certain players.
That's why I'm not recommending Chrono Trigger right off the bat, though that is largely an exception to the time travel problem.
For dungeon crawlers, (Like Pokemon Mystery Dungeon), I would recommend Etrian Odyssey. I didn't personally play this game, but it was well received.
For strategy, Valkyria Chronicles is a very difficult game, but its forgiving by the genre's standards.
I would recommend Fire Emblem games for strategy as well, but they typically don't age well price-wise. 13 was good and 14 is on the way, though the demeanor in 13 and 14 would probably be bad since its riddled with cliche and comes off something like a "high school" game.
There was also a Tactics Ogre game on PSP, but I haven't looked into it.
The key here is to start with something somewhat modern and work your way back. The quickest way to alienate someone from a genre is to start with its equivalent to The Pyramids. Sure, its a historical landmark they're historical landmarks that have stood the test of time, but building one isn't as pleasant as an IKEA chair.