Okay, here is a good variety ones you can look into (since you say you have played Minecraft together, I'm assuming that these recommendations don't necessarily need to be couch co-op, and by 'not graphic intensive' you mostly mean a game that can run on a computer that can run Minecraft on low settings)
BattleBlock Theater (platformer)
Keep Talking and Nobody Explodes (bomb defusing)
Hammerwatch (series of hack'n'slash games)
Don't Starve Together (survival)
Risk of Rain 2 (rogue-lite)
I have also seen Overcooked, Terraria, Portal 2, and Stardew Valley have already been recommended. Those are solid ones as well
Monster Hunter may be too much for your computers, but if you would be interested in a similar game that is lighter weight, you could look into God Eater (1 & 2 are bundled together)
Also, if you end up having any interest in branching out into vs games, you could look into Crawl, SpeedRunners, Screencheat, Ulimate Chicken Horse, and Lethal League
In no particular order and limiting it to 10:
Prince of Tennis OP 7 - Dream Believer: https://www.youtube.com/watch?v=7Uu7caB0OWI
The Unlimited Hyobu Kyosuke ED 2 - Brightest Light: https://www.youtube.com/watch?v=kHyRZY3Hb8s
Sousei no Onmyouji OP 2 - Re:Call: https://www.youtube.com/watch?v=blD1-ekFg9U
Samurai Flamenco OP 2 - Ai Ai Ai ni Utarete Bye Bye Bye: https://www.youtube.com/watch?v=Gdp4eGiU6Oc
Oshi no Ko ED 1 - Mephisto: https://www.youtube.com/watch?v=A48bcNyi1VM
Bakemonogatari OP 3 - Ambivalent World: https://www.youtube.com/watch?v=P0meri63JJ4
Bakuman ED 2 - Genjitsu to Iu na no Kaibutsu to Tatakau Mono Tachi: https://www.youtube.com/watch?v=C4BzBNmozKI
One Piece OP 2 - Believe: https://www.youtube.com/watch?v=54dp8ucsGG8
Bleach ED 17 - Hitohira no Hanabira: https://www.youtube.com/watch?v=MFTxGRE0BZI
Slayers Try ED 1 - Don't be Discouraged: https://www.youtube.com/watch?v=ByiqLpt6kKg