One thing I like to see in games is emotional responses other than anger, greed, fear and that heroic sense of great responsibility.
The Watch House, like Buffy itself when she fell for Angel, became more than a fun little Monster Of The Week series at around the point when the hero's player offered her up for heartbreak, by suggesting she fall in love with one of the other PCs. We made sure this was cool with the other player, and he further complicated matters by having him kick off a relationship with an NPC who could otherwise have been the hero's best friend. This emotional side ran through the game, creating a whole lot of angst and pain for the characters - which the players by and large gleefully participated in and acted to worsen, resulting in two of the PCs becoming seasonal Big Bads and more. It remained optional - a couple of the PCs were largely romantically untroubled throughout the six-year run of the game - but it informed a whole lot of the game.
Steve Darlington (who I hope isn't having to be evacuated from Brisbane as I type this) once wrote this article on Star Wars and it reflects my thinking on the setting well. One thing mentioned in passing in the section "The Characters Must Be Centre Stage to Everything" is the possible emotional weight of things. Granted, the films only really managed this in The Empire Strikes Back, but "No... I am your father" still packs a massive punch. I'd love to try a Star Wars game with moments like that as well as the more-often-done-in-Star-Wars-games moments of giant spaceships exploding and lightsabre battles. (The lightsabre battles in the original films are all big emotional scenes, something only Revenge Of The Sith really manages.)
I've played and run plenty of Vampire games over the years that didn't really touch on the Kindred's angst, pain, romance or much anything else except the political paranoia, and I was never entirely happy with them.
Of course, it's easy to lay it on too thick. I just watched Star Trek: Generations again last night, and killing Picard's family offscreen to make him feel a bit more vulnerable really sounds a bum note. (And not just because the Nexus as vaguely explained could have let him go back three days to save them, and phone Riker to stop Soren before the story gets started.)
Some games don't fit this. Some gamers don't want this. I wouldn't make it a big thing in an Indiana Jones style Adventure! game, for example, and the main emotional push of the Galactica-style military SF game I followed The Watch House with was "gloom".
Best to check in advance if you're going to do more than a little of this. The new Doctor Who is full of big emotional episodes, but the old wasn't so much, and there are plenty of classic fans who aren't happy about that...
But I wouldn't want to run a Buffy game, for example, that didn't have room for heartbreak as well as awesome fight scenes and one-liners.