Hmm... only five games? This could be tricky to narrow down, but doable.
First off though, some ground rules (some of which were mentioned by Extra Credits long ago):
1. Modern games only. Dated graphics are off-putting to those new to the medium, even though some age fairly well otherwise. A good rule of thumb would be anything from the current generation, though backwards-compatability of many of the current consoles does allow you to dip into the previous generation for some greats.
2. Something with a gentle learning curve, if not outright easy. Stone-walling players who are unfamilar with games with hard games is almost guaranteed to frustrate them and piss them off. Even though there are plenty of great games which thrive on hard difficulties, you don't want to alienate them.
3. Single-player only. Allowing the new player to go at their own pace will be better for them to begin with; competitive multiplayer is an ABSOLUTE no-no, and helping them out in co-op can make them feel inadaquate because you're doing all the work for them.
4. A decent storyline is beneficial, if not required. When someone says a game is "fun to play", it's usually biased; given as the view of someone who has played games for years, new players might have trouble seeing what you're seeing.
So without further ado... the list (in the order which things come to mind)!
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1. The Legend of Zelda: Twilight Princess (GC/Wii)
The reason this one is here is a combination of factors. First, it's available in two formats with different controls. The GC version uses a relatively standard control scheme, which could be easier to get into (though that could be the viewpoint of one who has played games with controllers for years already); the Wii version uses motion controls, and they work great in the game. Second, it's the newest Legend of Zelda title available (well, until Skyward Sword comes out in a couple weeks); a series with a track record of being some of the best games ever made, period. Third, it's the type of game everyone enjoys -- an epic adventure.
2. Portal 1 and/or 2 (PC)
Ah, Portal. The choice everyone is mentioning, possibly thanks to Extra Credits saying it's practically 90% tutorial. Very few games teach the player how they work so well, and through playing the game no less. The first game is a tight package, easy to get into and digest. The second has higher production values and is longer, but doesn't quite have the same level of challenge (outside of co-op mode, which is to be avoided due to the rules above) and charm of the first. Still, both are great games which anyone can play.
3. Super Mario Galaxy (Wii)
A bit light on the storyline, but the enjoyment of playing this is so pure that even those new to games will love it; this is a game which is meant to be played. It's so fun that I'm willing to waive the fourth rule for it. The first SMG is preferred here, for it's easier learning curve and slightly more serious tone (the second borders on silly). Anyhow, you can't go wrong with this one; it's been a fairly common suggestion already, and for good reason.
4. Final Fantasy IX (PS1)
At least one game where the story is front and centre had to be noted, which inherently implies the Final Fantasy series. Unfortunately, it's fallen from grace in the past few years... kinda forcing me to look at the older titles, but I can't go too old though. FFIX is pushing the age restriction a bit, but I think it's one of the best places to start; incidentally, it's the game which got me interested in the series (though not why I chose it). FFX was another candidate, especially with it's use of voice-acting. However, FFIX has that intangible charm to it; it also has a hearty dose of humour to it, which is less alienating than the darker storylines the other recent titles in the series have.
5. Shadow of the Colossus (PS2/PS3)
Another obvious choice, this game actually has a decent amount of tutorial-through-gameplay embedded in it. Every colossus is a puzzle, a mental challenge rather than one of coordination (which most action games are). The storyline is less explicit, with a lot of implied details and great atmosphere; just enough mystery to keep (new) players interested. As for the gameplay, it appeals to the fantasy of everyone's inner child -- taking down giant monsters. The act of awesomeness is muted by the sorrow & regret of having to slay each of the colossi, but that's just another layer of intrigue which urges the player onward. The recent HD remake helps make this game a tad more modern, despite it being essentially a timeless experience.
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Well, that's my suggestions at least. Most modern games, particularly big-name releases, lean towards those who already play games. Very few are as inviting to new players as the above. You don't need to get too heavy with dark themes and maturity; in fact, that's an easy way to push someone away from games in general.
You want light-hearted, fun, and easy-to-learn.
In other words, Nintendo.
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