Here are my suggestions. They should all be playable on your computer and are early 2000s vintage. You can find a lot of them for free here in the Internet Archive. I’m not sure how well most of these run on Linux Mint though, but a good way to check is to see if they’re playable on the Steam Deck since that’s a Linux based OS that’s popular.

Deus Ex 1 - Classic futuristic sci-fi FPS/RPG. Advanced technology, conspiracy theories, human augmentation. All wrapped in an awesome plot. Definitely try this one out if you haven’t yet. There are sequels, but none of them are as good as the original.

Unreal Tournament - Arena FPS that has excellent gunplay and runs on a potato. Can be played with somewhat decent AI as well. Has lots of well designed maps.

Half Life 1 - Classic FPS. It’s expansions, Half Life: Blue Shift and Half Life: Opposing Force are great as well.

Morrowind - a classic RPG set in the Elder Scrolls universe. Rich story and lots of ways to play it. Lots of mods out there as well.

SiN Gold - Action FPS with a rather bombastic plot.

No One Lives Forever - Funny espionage themed FPS with great dialog. Also has a sequel, No One Lives Forever 2, which has memorable lines like “oh no, more medical bills” when you shoot a henchman. You should be able to find these for free since their source code got released and was abandoned by their developers.

Soldier of Fortune - standard FPS where you play as a mercenary in various countries. Also has a sequel, Soldier of Fortune 2, which should be able to run on your computer.

Draken: Order of the Flame - Third person action game that was very much overlooked at its time. Had great on-foot combat mechanics and you could fly a dragon and engage in aerial combat. To this day there hasn’t been a game that really does this as well as this one.

Age of Wonders - Turn based strategy with a lot of depth in a fantasy setting. Its direct sequel Age of Wonders 2 should also be playable and has an excellent soundtrack.

Hearts of Iron - Real time 4x grand strategy set in WWII. I think its sequel, Hearts of Iron 2 is the best in the franchise and it has some expansions that add alternate timelines.

Diablo 1 - the original hack-and-slash isometric RPG. Also has a direct sequel Diablo 2 which should also be playable.

Neverwinter Nights - Isometric RPG, control a small grip of adventurers in a DnD setting.

SimCity 4 - A great city builder when Maxim was at its peak.

Need for Speed: Hot Pursuit - Racing game that puts you also against the cops.

Star Wars: Racer - Now this is podracing.

Descent: Freespace and Freespace 2 - Excellent plot driven space sims. There aren’t many of these out there, and these are some of the best in the genre.

Grand Theft Auto: San Andreas and Grand Theft Auto: Vice City. Both are great GTA games.

Mafia - third person story driven action set in the 1920s. A remaster was recently released last year, but the original should be playable.

Age of Empires 2 - classic RTS, the remaster should still be playable on your hardware as it isn’t too demanding.

Return to Castle Wolfenstein - ever wanted to fight techno-Nazis? Now you can in this FPS.

Assassin’s Creed 2 - one of the best in the entire franchise. Action third person, but without the RPG mechanics of the later entries.

Tom Clancy’s Splinter Cell - great this person stealth game. Has the best dynamic lighting of any game in its era.

Fear - horror FPS, has some jumpscares but also good gunplay.

Grim Fandango - A classic story driven adventure game set in the mythological Land of the Dead. This one runs fantastically on the Steam Deck so it is definitely playable on Linux.

StarCraft - another excellent RTS

Jedi Knight: Dark Forces 2 and Jedi Knight: Mysteries of the Sith - first/third person lightsaber action games. You might be able to run its sequels, Jedi Outcast and Jedi Academy as well.