These are my suggestions:
Not well known but seem to be well liked amongst those who've played them:
Breakdown (original XBox) - very good, but can be very hard at times,
Cold Winter (PS2) - pretty good, but is too easy,
Duke Nukem: Zero Hour (N64) - a really great game (my favourite Duke Nukem game), that is third person, but you can play in first person using a cheat, or an un-lockable option,
Singularity (PC, XBox 360, PS3) - mostly very good, it feels like a cross between Half-Life 2 and Bioshock, but the quality goes off towards the end, you can see that the game was probably rushed at the end to get it to the shelves,
Star Wars: Republic Commando (PC, original XBox) - I've not yet played this, but it seems to be very well thought of,
Unreal Championship 2 (original XBox), a really good game, in some ways the best Unreal game, but it should have been ported to the PC so PC gamers could get to play it, since the other Unreal games were always on the PC.
First person shooters that I consider to be under-rated:
The World is Not Enough (N64) - most people dismiss this as in inferior Goldeneye clone, which it is, but it does a lot right in itself, and is a lot of fun,
Timeshift (PC, XBox 360, PS3) - *
Turok 3 (N64)
* I think Timshift is the worst game in this post. It does a few things amazingly well (such as the time powers, the weather effects, and some good weapons), but it went through 'development Hell', and it shows. The game is very uneven, with the good parts being mixed with bad or boring level design, sudden deaths that you can't help failing at the first time you encounter them, the best (most enjoyable to fight) enemies only come near the end so you don't get to fight them often, the story is extremely badly told, the ending answers *nothing*, and it just feelings unsatisfying.
I mean, I've completed it several times (because when it's good, it's really good and enjoyable), but I still don't know much of the plot, or what the ending is supposed to be mean. The game's development went though two major changes of direction, and in the final one, which is basically the game that we were given, they stripped away so much of the plot and the characters' backstories and whatever, that it's difficult to know much of what was intended to be portrayed by the story.
The character you play, a scientist turned first class gunman/survivalist, has no name (that we're ever told of), and we know nothing about him. But it's hinted that he was involved with another (female) scientist, and there's obviously some sort of power-play driven intrigue behind the theft that starts the game. At the start of the game, in modern times (say 2008, around the time the game was released) in a large scientific research centre, one of the higher up scientists, called Dr Krone, steals a new invention, a time-suit (the Alpha Suit), which allows the wearer to travel through and control time. Before he time-jumps away, he activates a bomb in the building.
The hero of the game manages to get to, and don, another time-suit, and also jumps just before the bomb destroys the building and kills everyone in it. The hero appears around 1939, but it's not the 1939 of his (our) past. The villainous scientist, Dr Krone is now some sort of dictator, controlling some or all of Earth (exactly how much of the world is ruled by him is only one of hundreds of things that the game never tells us), as Dr Krone has changed history. This 1939 now is now evidently mostly fascist ruled, and with technology *far* in advance of what it should be, and in some ways more advanced than the 2008 when both scientists jumped back in time.
The hero of the game has the 'Beta Suit' (as opposed to Dr Krone's Alpha Suit), but it's never made clear what the differences are, and though presumably the hero's beta suit can jump through time (since it just did), you never get to use this feature yourself, instead you get to pause, slow down, or rewind time at will, but only for at most a dozen or so seconds at a time. The suit also generates a shield to protect itself (and so the user, too) from physical harm, and it covers the who of the user (including their face, so no one looking in can see the user's face).
The world of 1939 where you start the game, contains an organised, armed rebellion against Dr Krone's much better armed forces, and when you arrive, because this is a game with little attempt made to fix any plot holes, then the rebels trust you implicitly for no reason at all. The rebels don't know who you are or where you're from, you're wearing a strange armoured suit that they've never seen before, and they're facing certain death from all sorts of much better equipped and more numerous and varied official military personnel and machinery, but none of them even thinks "This stranger we've never seen before, in a strange suit, who hasn't even shown us his face, could he possible, just possibly be an enemy, maybe a spy sent to infiltrate us?".
Throughout the game you constantly encounter new rebels and new official soldiers, and the former always trust you straight away, and the latter
seem to know that you're an enemy. Alright, so maybe Dr Krone told the soldiers that "If you see someone in a suit that looks like this, then he's an enemy, so kill him", referring to your time-suit, but it doesn't explain why the rebels overcome both their paranoia and common sense to trust someone they don't know.
But game-play-wise, there is so much potential here - you can slowdown, pause, or even take back time whenever you like, and if works amazingly well. Whenever any of these powers are active, then you can, if you choose, still move and shoot at normal speed, so if you pause the game, and shoot some (now frozen) enemies with your shotgun, then they won't move (because time is standing still) but when time restarts, then they get thrown back and killed by the blasts. But it's mixed in with boring, sometimes even tedious level design, and even though the game seems professional (no bugs, the graphics are great, the backgrounds and art style do a great job of portraying a not-exactly-the-past-of-our-Earth alternative timeline, and there is so much to like about the game. But then you arrive at one of the boring parts, and remember why the game got such bad reviews.
The final nail it's coffin is that it's non-moddable. A real shame, as good modders could really have gone through the game, altering parts of some levels, placing more enemies in different places, etc, and if they were really good then they could have modded in a complete story (even if it was only in the relatively extremely simple method of adding story-based text in the loading screens between levels, and even added upgrade abilities to you time-suit and weapons abilities. The game is long, and collecting upgrade points to improve your defensive and offensive abilities would have been great.
Oh well. Still, it's very cheap now, should hopefully run on any PC, and if you like science ficton first person shooters that aren't great but do have flashes of brilliance then I would say try Timeshift.