A damn hard choice. Damn hard.
Half-Life/Team Fortress/Counter-Strike
From all technicality, this is a subrace of the Quake line. Team Fortress started as a Quake mod, Half-Life was built on the Quake II engine, and of course Counter-Strike was built on Half-Life. However, the modifications that Valve made to the engine brought it into it's own species. The brilliant design choices and excellent pacing, as well as Valve developing the Source engine, which, even when it's 7 years old still stands up to modern day graphics from the Unreal engine, This was going to be the choice for me. That until I saw number two.
System Shock/Deus Ex/BioShock
You can also add in the Thief series here too. Mentioned earlier, System Shock 2 and Deus Ex are both regarded as some of the best games that have ever been made. The freedom, the choice, and the ability to customise your characters without being penalised for going a certain route is probably the one mechanic that the majority of games should follow. System Shock 2 stands as one of the most atmospheric and downright thrilling games I've played, even with the outdated graphic (the Rebirth Mod and SHTUP have helped alleviate this somewhat) and even Bioware haven't made a game that beats the epic scale of Deus Ex.
This where my vote went, even if BioShock was just a little too similar to System Shock 2 for my tastes.
Wolfenstein/DOOM/Quake
Pioneers of the genre, iD has spawned one of the biggest genres in gaming. Likened somewhat to Black Sabbath, they invented the genre. They've done nearly everything in the gameplay. Sure you may be doing it faster, slower, with a story, or what have you, but they've done it all. And after the release of Quake III, they cemented their place in history, with the Quake II engine fueling some of the greatest games the genre has seen, even to the point where they actually made a Star Trek game that was actually good. Now that's an achievement. It's a shame that they haven't done much since then. Well, not much decent anyway.
GoldenEye/Perfect Dark/TimeSplitters
A lot of people said that Bungie (and by proxy, Halo) are responsible for the console FPS. These games, however, are the true pioneers, proving that you could indeed get the same amount of fidelity from a simple analog controller, whilst still making them just as fun as anything else that the PC had as an FPS. Granted, I only played TimeSplitters, but most people have got fond memories of GoldenEye and Perfect Dark.
Call of Duty
Call of Duty, oh Call of Duty. CoD have simply set trends, and then made new ones. However, the main problem I have is that sheer amount of money that they make is encouraging a metric fuckton of mediocre knock-offs. Modern Warfare had one of the best scenes in it, (you know, The Nuke) but all I can say, is that this series doesn't really have anywhere else to go.
Duke Nukem 3D/Blood/Shadow Warrior
There were two sides back in the mid '90s. You were either a Quake fanboy, or a Duke Nukem 3D fanboy. The antithesis to the dark and creepy Quake; Duke, Blood, and the other Build engine games had a sense of humour, which is something that is really missing from FPS games these days. Bulletstorm was close, but not quite to the same level. Duke Nukem Forever is something I'm curious about though.
As a side note: More games you can add to this linage are the games made by Monolith, who took the ideas from the Build engine and made their own, namely Lithtech. Alien Versus Predator, F.E.A.R. the serious offspring of these games.
Unreal
The only competitor to the Quake engine, and once again spawned another fanboy war, namely between Unreal Tournament and Quake III. Epic made one of modern gaming's greatest assets these days: The Unreal Engine (We're up to the Third Generation now) which has given developers in all genres the tools of making awesome looking games, but it's a shame that they haven't really made anything with the same fun factor of Unreal, or Unreal Tournament.
*sigh* I just miss the RazorJack.
Marathon/Halo
Halo codified in stone what the formula for FPS needs to be. Space marines, regenerating health, and controls that any one can pick up and be good at. Not everything is bad here, but I'm just wishing that Bungie would hurry up and make Oni 2 already.
There was a couple you missed too:
Battlefield
The closest thing that the FPS has to the MMO format, (MAG tried, and didn't get that far) but Battlefield has always had really decent, solid game mechanics, even if the series was a bit buggy. This series also beat Call of Duty to the 'Modern Warfare' title, with the Desert Combat Mod for Battlefield 1942. So many wasted hours there.
Starsiege: Tribes/Tribes 2
Formative in my early FPS days, this was one of the greatest team basted FPS until the release of Team Fortress Classic. With an almost limitless skill ceiling, Tribes had a similar (read: almost identical) design to Halo, but was just released a couple of years too early, combined with the fact that it was before the broadband revolution. Revered by fans, and ignored by every one else. Which is a shame.
I've really got too much time on my hands.