Battle For Wesnoth is an open source turn-based strategy game with multiplayer that will prolly run on anything. It seemed pretty good, I only played it once tho

Sven Coop is good shit albeit an FPS. It's a free coop mod of HL1, you can play through the entire half-life campaign in coop, and there's like fuckin hundreds of other coop maps people have made in like any genre, you got puzzle maps, you got escape room shit, you got mario party maps, you have a faithful recreation of the first dungeon of Ocarina of Time. You name it and someone's probably made a hl1 engine remake of it

War Selection is an RTS online battle royale, i dunno how low-spec friendly it is and i only ever played it for like five minutes

Not free, but Dominions and Conquest of Elysium are pretty fun coop games in the grand-ish strategy genre. Dominions is more of a traditional grand strategy game with nation management, conquest of elysium is more of a lower scale band of heroes type thing by the same devs. I highly, highly recommend pirating them unless you're strapped/they're on sale tho, as they are definitely not worth the like 40 bucks they ask for imo (multiplayer worked fine on a pirated copy). Good stuff tho, tons of different factions with varied playstyles. I won a game by playing as a undead army that kept on spamming a global rapid aging spell that didn't affect my skeletons, so that was pretty rad. edit: kinda a big learning curve tho now that i think about it, the UI was absolute ass

There's tons of online/co-op text adventure games on the internet called MUDs, AI Dungeon allows for multiplayer text adventure stuff, but I think that feature has a paywall and if anyone puts a horny command into the chat once the AI will start nosediving into a thirsthole (which can be hilarious). Space Station 13 is essentially a text-based game with minor graphical elements and can be a blast, but the learning curve is astronomical and some of the communities can be pretty shitty