My kid is two and enjoys watching (for short periods of time) Commander Keen 4 and the Sonic 2 port I got as part of a Sega Genesis Collection on Steam. I find that something with fun music also helps, and Jazz Jackrabbit qualifies for that even though it can be infuriating to play at times.

If you want them to get into a fighting game that's not as violent as Mortal Kombat or others, One Must Fall 2097 is a great choice. It can also play a bit more slowly than other fighters, making it more forgiving for a kiddo who's still learning how to play (or for old parents/aunts/uncles/grandparents to play along).

James Pond 2: Codename: RoboCod was something that kids really liked playing. When some relatives were visiting they played it pretty much all weekend until they managed to finish it.

We used to play Unreal Tournament with my little sister when she was 8. She loved sniping people, running to where they were, laughing at them, and then running back to her computer and sniping them again.

We allow then only non-violent games.
- Croc was already mentioned.
- Pandemonium 1 & 2
- Midtown madness
- The Pink Panther: Pinkadelic Pursuit

All of this runs on an old Pentium M laptop with an ATI X600. Bought 7€ on ebay for that purpose

Stargunner and Tyrian are also great if you like scrolling shooters. Both with great sound tracks.

How about playing catch with the old man instead of more screen time?
Just kidding. My 8 yo loved Aquanoid and Pharao's Tomb and I'm about to introduce her to the SNES emulator so we can game together.

Thanks for all the suggestions!

I saw Freddi the Fish listed and I was like...wait...wasn't that an old CGA came. Nope! Apparently two different games Freddy Fish
Freddy Fish https://www.mobygames.com/game/13097/freddy-fish/
Freddi the Fish Series https://www.mobygames.com/group/479/freddi-fish-series/

And WRT screen time, my kids actually get very little, so they aren't completely spoiled yet! 😀

https://www.youtube.com/watch?v=9jAlEQ-VZaw
My niece used to love this. Probably still would.
Not sure if she actually played the game but she loved the music.

Also, "The Learning Company" made some good DOS games. One I like is Operation Neptune.
It seems prone to lock up on the intro screen, but it works on my 486. Maybe it's speed sensitive?

My boys, ages 8 and 11, tend to like:

MS-DOS:
Wolfenstein 3D
Duke Nukem 3D
Super Street Fighter 2 Turbo
Quake

they tend to like any fighter or shooter game I hand them. This is all I've handed to them so far. Games are used as reinforces for good behaviour in our house. There's no free-for-all game play here.

If you are considering NES, I recently pulled my Nintendo stuff from the attic. My boys liked:

NES:
Contra (but only with lives code, otherwise they get angry)
Super Contra (but only with lives code, otherwise they get angry)
Legend of Zelda
LOLO
Solomon's Key
Double Dragon

Super Mario Brothers makes them angry as they die too easily. Same issue with Tetris. They, surprisingly, did not like MegaMan 2. I remeber enjoying Contra in the 80's. It lets you play two players simultaneously. I also recall enjoying Zelda and Super Mario Bros 3 from the 80's.

My daughter, age 13, seems to have lost all interest in my vintage games. However, be warned, there's this heavily addictive iPAD game called Roadblocks that turns children into vegetables. Take away the Roadblocks and your children will murder you. Whatever you do, don't let them play these modern iPAD games. You will regret it. I have to keep the iPADs locked away.

My take on this...

For younger kids (7 and under), give them some simpler games so they can learn how to use the keyboard and mouse (separately.)

I find Blocks (the one where you have to flip and find two matching blocks) to be good for getting kids to learn how to navigate around with a mouse.
Also, let them draw with MS Paint for the same reason.
Don't forget DXBall! 😀

In regards to learning the "baisc" keys on a keyboard (i.e space bar, arrows, ctrl, alt, etc.), I think Commander Keen is pretty good for that. And it's an outstanding game, really.
But even if you don't have a proper retro PC to run something like Commander Keen (or DOSbox setup), then something as simple the Pinball game included with Windows XP can still provide hours of fun. In fact, I just showed that one to one of my nephews last week, and he was totally hypnotized by it (which was quite the exception, as he doesn't care about any other video games, especially on a phone or tablet.)

While at the "learning how to use keyboard" stage, I find that racing games can also be quite fun for most kids. Any of the retro Need For Speed series should do here (except maybe Porsche Unleashed / Porsche 2000 - it's a really fine game, but most kids just won't appreciate the amount of understeer in the game's physics.) I'd say NFS 3/4 would be the easiest to pickup... not to mention many of the maps have jumps and other quirks that almost always make for a unique driving experience. They are hard games to master, though.

Another one that my niece and nephews really liked once they got a little more comfortable with KB & MS is Portal - yes, the one from 2007. I suppose one could argue it's not that much of a retro game... but it's still a fun puzzle solving game in 3D space. The sequel, Portal 2, is also very good.
And if they want to try a more "sandbox" type of game, but with decent real-world physics, give them Garry's Mod.

Once the kids are more used to navigating in 3D space with keyboard and mouse (and I'd say when they're older than 8-11 years), certainly try any of the popular multiplayer shooters - i.e. Quake (1/2/3), Half-Life Deathmatch, Counter-Strike (1.5/1.6 or Source), and etc.

As for Half-Life (single player mode) - save that one for when they are older (I'd say >11-13 YO). It's a great game... but boy, I still remember the shivers it gave me the first a friend and I played it. Then again, it probably didn't help (or maybe that's what did it) that we were playing it on a CRT, late at night, with no lights in the room, and our parents "not around".

I started my kids out with Ms. Pacman in DOS.

.. as a kid I remember having fun playing In Search of Dr. Riptide.
It was fun playing the remote-controlled robot sub. Tails' Adventure uses a similar concept (there's an SMS port now).

Other underwater games are EcoQuest: The Search for Cetus, Ecco the Dolphin or SeaQuest DSV (consoles).

Whacky Wheels also was fun. PC version of Super Mario Kart.
Hm. Now that I think of it, some of those games are already shown in my "320x200 256c games on a CRT" thread. Strange. I didn't realize I was that kids friendly, hah. 😅
MCGA Games (PC/DOS) - LCD vs CRT

Edit: There also were fun and non-violent point&click games such as Mission Supernova or the many advertisement games.
Some edutainment games were made by federal agencies,
to make kids aware of environmental protection or prevent hostility towards foreigners etc.
Unfortunately, these game's aren’t available in English language.

The closest that comes to mind right now would be Inherit the Earth: Quest for the Orb.
I can't help but it always reminds me a tiny bit of an anthromorpic version of The Animals of Farthing Wood.
But that's probably just me. 😅 I guess I watched too many cartoons at the time.
Speaking of, it makes me wish there was a game based on The Dreamstone..

Thanks everyone for the suggestions. I have compiled the games into an Excel spreadsheet where you can organize by Name, Year, etc. See the first post in this thread for the file.

Seems like there were thousands of DOS learning games with low system requirements of all kinds even freeware that the under 7 group would play. Most are probably lost to time.

https://m.youtube.com/watch?v=BlFnGgZHPzY