I love the original Xbox. I think it's up there in the high tier along with the NES and SNES. I've spent years looking for the best games on the system, both critically acclaimed and remembered and the hidden gems that you need to dig for, but once you unearth are pure gaming bliss.
Here's my recommendation list.
Call of the Cthulhu: Dark Corners of the Earth - A moody, atmospheric horror game based on Lovecraft's classic novels. The game looks great and is interesting to play, but also looks slightly aged only because its not high def. This is technincally an FPS, but you shoot in it about as often as you stand still in a 2D Sonic game.
Castlevania Curse of Darkness - A sequel to Castlevania III, This is probably the only somewhat decent 3D Castlevania. It's still not nearly as good as the 2D Metroidvanias, but its nevertheless a worthy title of the Castlevania name. The Xbox version looks better than the PS2, surprise surprise.
Chronicles of Riddick: Escape from Butcher Bay - Even if you don't like the film or Vin Diesel in general, this is one awesome game. Stealth action, great graphics, well aged.
Conker: Live and Reloaded - A remake of the N64 classic. Pretty much the same as that game, so you'll like it or hate it depending what you thought of Bad Furday.
Dead or Alive 2 Ultimate - I consider this to be the best fighting game on Xbox. Released well after Dead or Alive 3 and all previous versions of Dead or Alive 2, DOA2U looks damn impressive, even DOA4 doesn't look much better. There are 20 costumes for all the chicks and about 10 or so for the guys, its fun, DOA chicks are hot, and it also comes with the Saturn version of Dead or Alive 1 which is fun for about 10 minutes. You can also pick it up for around $6. GET IT.
Doom 3 - If you somehow missed Doom 3......get Doom 3 Limited Edition which you can find for about $10.
Doom 3 Resurrection of Evil - Moar Doom 3 love, and the balls to the walls hard Master Levels of Doom 2.
Dreamfall: The Longest Journey - A rare example of a console Adventure game that's not a port or a remake of something from the mid 90s, Dreamfall is the sequel to the 1999 PC game of the year, The Longest Journey. Dreamfall is an amazing experience that has some flaws tacked on to make it more console gamer friendly, including largely useless stealth sequences and awful combat. Still, this game has some of the best voice acting EVER, an incredibly engrossing storyline graphics as good as the Xbox can produce and a wonderful score. Play through it and join me and the rest of the Longest Journey fan community that are pissed off of the third title's delays.
Fatal Frame - Rare, expensive, and worth tracking down, Fatal Frame is in many ways Japanese Resident Evil, which evil Japanese ghosts and curses replacing zombies. The gameplay consists of taking good photos of ghosts to sap them of their energy and ultimately unravel what happened in the mansion you're in. Still not convinced? You play as a cute underage Japanese girl in a school uniform. If that doesn't make you want to get it, nothing will.
Fatal Frame 2: Crimson Butterfly - FF2 is similar to the first game, but its longer and has a greater emphasis on puzzles. Tecmo went a little crazy for this one with the main characters being not one, but TWO underaged schoolgirls. The Xbox version includes the same third person game from the PS2 and also a new first person mode that sort of makes it Fatal Frame the FPS. This mode doesn't really work as well as was dropped in the next two games.
Heroes of the Pacific - A WWII Flightsim that's really cool and a lot of fun. If you're a war buff sick of WWII games set in Europe and requiring you to load up your M1 and Thompson, give this one a shot. It's lengthy and very fun. Two sets of controls so anyone can enjoy it regardless if they want arcade action or complex flight sim controls.
Indigo Prophecy - It came before Heavy Rain and shares a lot in common. Not so much a game as an interactive movie. It's not really my thing, but its an interesting game worthy checking out.
Jet Set Radio Future - An update to Jet Grind Radio for Dreamcast, this game is even more amazingly fun and features some of the best damn music in any game.
King of Fighters NeoWave - I consider this KOF 2004, this was the first King of Fighters game made or arcades to not be on the Neo-Geo MVS. NeoWave is a beautiful 2D fighter featuring a giant cast of characters and the Xbox port is handled very well. I don't like it quite as much as SVC Chaos, but it's a great game just the same.
Malice - The final game by Argonaut Software (the folks behind the original StarFox) Malice is a fairly generic 3D platformer that nobody besides me seems to like. The game is quarky and a bit rough, but its not nearly as bad as most would have you think.
Marvel Vs Capcom 2 - It's gonna take you for a ride. Port of the Dreamcast version. No online support, which apparently renders one of the best fighting games of all time a 4.0 according to IGN. Don't be fooled: this is the definitive version of the game in its original form. The XBLA version revamped the menus and options and took out unlockable artwork. This is how me and my buddies prefer to play this classic game.
Metal Wolf Chaos - You play as the President of the goddamn United States in a mech defending American freedom and justice from the traitorous vice president who also rides a mech. Released only in Japan, this game does make one wonder how the Japanese think the US Political system works. The game is completely dubbed in English with only Japanese subtitles. Fully playable and totally awesome. This is the most patriotic game of all time.
Ninja Gaiden BLACK - Do you like Ninjas? Do you like HARD games? Do you like the fact that the Xbox is too big and heavy to easy throw across the room in anger? Then this game is for you!
Otogi 1 and 2 - The Otogi games are beautiful Sega made action games that sold poorly due to obscure names and heavy emphasis on Japanese mythology, but goddamn do these games rock. You might be able to luck out and find them cheap.
Panzer Dragoon Orta - It's like the Saturn classic but its prettier.
Petit Copter - A friend living in Japan managed to buy this game for 500 yen but spent nearly double that shipping it to me (lol) but yeah. This game is a fun remote controlled helicopter sim released only in Japan.
Psychonauts - One of the best 3D platformers I've played. I'm sure someone else can explain why its so great, but just trust me it is.
The Red Star - The Xbox version was finished, canceled, and leaked online, and is superior to the PS2 version. The game is a mix of two awesome oldschool genres mixed together: the beat 'em up and the bullethell shooter. It's impressively done and is a true hidden gem on the Xbox. To play it you of course need to legally buy the PS2 version and then softmod your Xbox to run backups. The game loads quicker than the PS2 and of course looks better to boot.
Return to Castle Wolfenstien - A wonderful successor to Wolfenstien 3D, this version is better than the PS2 and has more content than the PC version. I think its a better game than the 2009 Wolfenstien.
SERIOUS SAM - Superior Xbox version of The First AND Second Encounter. Also cheaper than the XBLA release of only one of those, let alone both.
SNK Vs Capcom SVC Chaos - One of the final games made for the Neo-Geo MVS, SVC Chaos is a masterfully crafted 2D fighter featuring fantastic original sprites for all Street Fighter characters and features some hidden awesome fighters like Zero from Megaman Zero and you can play as a Mars Person from Metal Slug. I mean come on. How can you NOT love it?
SoulCalibur 2 - Still good after all these years.
Taito Legends - A Collection of many awesome classic Taito games including Bubble Bobble, Elevator Action and Space Invaders. The Japanese version includes Pu-Li-Ru-La, and that game is so overly bizarre it's worth the import alone.
Taito Legends 2 - Released only in Europe and Japan, this awesome collection includes some of the best arcade games ever such as Elevator Action Returns, Puzzle Bobble 2, Bubble Symphony, Gekirindan and Growl. It also has The Legend of Kage, which arguably hurts the value of the package.
Tenerezza - A cell shaded, anime action/adventure RPG game somewhat like Zelda. Tenerezza is a young anime witch-girl...you'll need a guide to play it but its one of the most interesting Japan only games.
XIII - A highly stylized FPS based on a french comic nobody outside of Europe has heard of. It's a downright SHAME no sequel has been made yet.
There are probably more, but man this took way too much time to write.