I always joke with a friend of mine that I've seen his now 16-year-old play Farming Simulator on Steam, again. And that something obviously went wrong there. If done right, the boy would've been raised on Doom, Quake and Painkiller from the start. Again, it's a joke we can both laugh about, and the kid should play whatever he likes of course.

Needless to say, I'd never let a child near a really violent or potentially psychologically disturbing game.

Here are a few recommendations:

Hammerwatch
Startopia
Disciples: Sacred Lands Gold ;the whole series maybe
Volgarr the Viking ;kids need to get good ;)

I strongly recommend Webbed and Yoku's Island Express.

Dragon Cave

It's a free dragon collectables game where you have to catch, breed, and raise dragons. No fighting or violence involved. But you have to take care of your virtual pets - make sure the dragon egg hatches safely, then raise the hatchling to adulthood.

You can name the hatchlings when they hatch, and then breed adult dragons. I really recommend it. If you're really lucky, you might catch a chicken, paper dragon, cheese dragon, or a dino.

Not on GOG, but definitely Untitled Goose Game.

Puzzle Agent is great.

Botanicula is cool.

World of Goo is awesome.

When I was a kid, I really enjoyed the 18 Wheels of Steel games, Across America in particular.

Army Men series.

Rollercoaster Tycoon series.

Then again, kids these days are probably not interested in games I played nearly twenty years ago, but there they are.

Kids are just smaller adults who are developing constantly. While it is understandable to feel the obligation to keep them out of things they don't understand, in terms of video games, establishing what is fantasy and what is reality is an important baseline. As a parent, you should be there to take any question they should ask about a game.

I'd start with Avernum, as I sank hours into Exile when I was a kid.
Epic Pinball, but maybe null out the Cyborgirl table. (It's junk anyway.)
Re-Volt, even if RC cars are a largely bygone thing, is great and can be extended forever thanks to source ports.
While Cave Story isn't on GOG, I think it is evergreen enough to provide entertainment to all. Just don't get a Nicalis published version. They did to Pixel what Elorg did to Alexy.
Speeaking of hi-fi games, OpenTTD is entirely benign and could help teach some basic concepts.
And if you want your child to be deeper thinking, Myst is a great series. Just maybe help them take notes.
Kids played Harvest Moon, so Stardew Vally (especially if you can swing local multiplayer) will be a fine time.

Hello Kitty always works for my childrens.

The RollerCoaster Tycoon games are a MUST! Not only are they fun because of the ability to design and operate your own amusement parks but they also teach some valuable lessons that could potentially help kids learn the basics of economics.

Depending on how old they are, Spiritfarer is a very relaxing game with some deep themes. Probably better for kids who are nearing their teen years, as the themes delve into concepts like the afterlife. Then, of course, there are games like Terraria or Stardew Valley that'd be kid-friendly as well.

Skyrim wouldn't be bad outside of some minor bad language and mostly non-graphic violence but the parent would need to make sure their kid doesn't enable perks like the one that allows you to behead your enemies. Also probably best to make sure not to tell the kid about ALL of the freedom you can have by playing the game, as just about every kid who plays TES and knows about its freedom goes on a killing spree at some point lol

The Myst games are completely calm, but I'm pretty sure kids wouldn't enjoy them because they have notoriously difficult puzzles. Even so, it might not be a bad game for parents to play WITH their kid, especially if the parent enjoys difficult puzzles.

Most Star Wars games would be kid-friendly but the most kid-friendly ones are probably the Lego Star Wars games, as well as Episode I: Racer, Rogue Squadron 3D and maybe the Jedi Knight games (just be aware that Jedi Outcast has some confusing level design).

If the kid can read and has a little bit of previous experience with faster-paced action games, Tokyo Xanadu eX+ is a great JRPG that's about school kids fighting monsters (ones that aren't particularly creepy) but it's also deep enough for adults to enjoy.

Spongebob Squarepants: Battle for Bikini Bottom Rehydrated is DEFINITELY something kids can play. The original was a game I actually enjoyed as a kid, myself, and playing the Rehydrated version just brings back so much nostalgia. Thing is, it's a very well-made game too so those kids who play it may end up enjoying it later down the road.

As a kid, I also played a LOT of the Total Annihilation games. Both the original and TA: Kingdoms are great games! They're real-time strategy games but they're simple enough to learn and can be played on virtually any system, through LAN, which is what my own dad used to do when I played it as a kid.

Also, Sid Meier's Pirates! can be quite a bit of fun! The original (listed here as Pirates! Gold Plus) was a game I played on my grandmother's old Compaq PC and it was my introduction to open world games, which is one of my favorite genres today. Kids would probably prefer the newer version due to its updated graphics and better gameplay but both are great.

Finally, I'd say that if the kid is a bit older, the Civilization and SimCity games are great. Both are probably far too complex for a younger kid but for ones that are approaching their teens, both games are a lot of fun and can help teach lessons similar to how RCT does. Civ can teach a basic understanding of how supply-and-demand works, how international diplomacy works and so forth. Meanwhile, SimCity can help kids learn the basics of how cities are designed and how much work goes into city planning... and hey, if they get bored with that, they can always build a city and throw a bunch of disasters at it!

Lost in Play is a recent game. And it fits perfectly for kids : cartoon visuals, non-violent game, simple but logic puzzles. It's a Point'n'Click game, so it's a good start for them !

I wouldn't recommend any digital games at all to kids. Gift them some board games or Lego sets - Lego as name for the building bricks/system and not the corporation.

All the LEGO games, definitely.

I'm planning to make my kids play through the entire history of video games, in chronological order. They're on Pong at the moment - might be ready for Ultima 1 in 2-3 years.

Do they have to go through the ET game as well?

Edit: And don't forget to grab an Amiga or at least a good emulator. That thing had tons of great games, even after the console market went down..