I'm a big fan of the first four Splinter Cell games. They're well known for their good third-person single player modes and for encouraging the use of a smarter playstyle that maintains a greater focus on being stealthy and undetected as much as possible rather than going all rambo and run n' gun.
I'd recommend the fourth game, Splinter Cell Double Agent. It's the only one of the four that I know for sure is available on XBox 360. The fifth and sixth games, Conviction and Blacklist, are on XBox 360 as well, and as individual games, they're not bad and pretty fun, though I personally dislike them as Splinter Cell games as they are far, far too "shooter-friendly" in my opinion, and I feel that the development team behind them did an awful disservice to the character of the series' protagonist, in making the decision to take the franchise in the direction they did. Though I'd still recommend them if you enjoy third person shooters otherwise.
Double Agent is good because while it lacks a more believable, politically focused storyline and is notably easier than the first three due to Tom Clancy fully giving over the game's rights to Ubisoft some time between the creation of the third and fourth games, its storyline is more character focused and story driven, giving it a more adventurous feel like you prefer and allowing for a slightly more cinematic feel that doesn't try to go too far over the top to be a "movie" like Conviction, Blacklist and too many other games today try to.
It was released in 2006 as one of the first XBox 360 games on the market, and its graphics are still fantastic for the system it was on and I'm fairly sure they even hold up pretty well in comparison to some current gen games.