I enjoyed tabletop role-playing games as I was growing up, but I wasn't as young as 7-8 when I started. Are there any good ones that you could recommend for me to play with my son and his friends?
In general, almost any game should work at that age, you just have to be prepared to adapt. Plenty of 8 year old kids have picked up their older sibling's D&D book and played with their friends, having a lot of fun even if they're not calculating Base Attack Bonus properly. Kids are imaginative, so I'd try and encourage roleplaying and worry less about the pure mechanics of rolling dice, which is easy to screw up.
Biggest advice: make sure you understand the magic system well. Even among adults I've seen confusion about how exactly spells work (for instance in Beyond the Wall, there's cantrips which always work and spells which have a % failure chance, and that trips people up)
Short self-contained adventures with a simple ruleset
If you're looking for something more "storytelling-focused" rather than dungeon-crawls-and-monster-fighting, there's another good child-focused tabletop game called The Princes' Kingdom by Clinton R. Nixon, which in turn is a hack of Dogs in the Vineyard by Vincent Baker. It's about the young rulers of an island kingdom travelling around from island to island and solving their citizens' problems, and has a really interesting conflict resolution system it borrows from Dogs which involves rolling a bunch of dice up-front and then using the rolled as resources in a back-and-forth conflict: https://www.rpg.net/reviews/archive/12/12593.phtml
You could also try some board-game RPG's instead.
But overall it went great, maybe too great, haha. Afterwards they'd frequently try to suggest that instead of any normal camp activities they could play D&D. The camp basically made a rule against it because too many kids got interested and you'd either have an unmanageably large game, or have to exclude kids from the game.
I think the 5e starter box is somewhat similar. I know it comes with a reduced rule soft-book with just the important rules and spells. I believe it includes premade characters too but I'm not 100% sure.
If you go this route, ask for what kind of character they want to play, create/find a pre-gen, and go from there (don't bog them down with character creation, -especially- if they want to be a spell caster). Don't worry about going into depth in the rule system beforehand, let it come up organically. Try to get a game with friends first so you know the rules, so you don't have to slow down to look things up, and be willing to adjudicate freely (as kids, especially, may want to do very unorthodox things).
All the usual DM/GMing rules apply. Make failures as spectacular as successes. Try to describe things in ways that are interesting, rather than just "you hit it for 8 damage". See if the kids engage. If they do, they'll pick up the rules as you play. If not, move on.
With that said, 5th edition DnD is pretty easy to get into, and it is a lot of fun!
The 3 adventures it comes with are a little weak IMHO, although that may just be my poor story telling ability. The ones in Little Wizards look better, but I haven't had a chance to try those out yet.
Dice and systems should not get in the way. Just teach them the very basic (what is passing/failing a dice roll) and add nuance from there.
I'd suggest finding a youtube tutorial though, the rulebook is unnecessarily verbose and confusing.
edit: I haven't played Munchkin in 10 years, so they may have updated the rules somewhat.
Some of the variations can be fun to play (they add interesting complications to it). But only in small doses for me.
Now that we're talking about this, I'm curious of any good FATE games to try, particularly for a 2-player group as we try to learn the system.
Way simpler than D&D, and has a lot more humor built-in. You can get pretty much all the rules and stuff for free, online.
+ Pure D20
+ Slim down D&D 5e and play it a bit fast and loose
I remember playing "D&D" with my dad at about that age. He used a D6, but basically he used a simplified D20 system with a lot of fudged rolls and minimal mechanics.
Fate Core is great at just telling a really good action adventure without concepts of inventory or death. It has some optional mechanics to let the players add to the story (recognizing the person who has kidnapped them, having been a smuggler in this town before, etc).
If your players enjoy jenga and horror, you can always run [Dread](https://dreadthegame.wordpress.com/about-dread-the-game/) which is very fast to get running and runs very smoothly, even for folks who haven't played an RPG before.
Oh, and finally, game sessions last only about 15 minutes, which fits within the attention span of children.
D&D 5th Edition does streamline a lot of things compared to 3rd and 4th edition. It also brings D&D (to some extent) back to its roots (your characters aren't veritable superheroes out the gate like a lot of them seem to be in 4th, for instance, and you aren't selecting from tons of classes across the various splat books like in 3rd).
Castles & Crusades is a D&D-like system that's fairly light and meant to call back to 1st and 2nd edition. No skills, no feats. No THAC0-like complexity in the rules. You have a target number to hit based on your primary stats, and modifiers based on the situation and abilities. A fair number of published modules of decent to good quality and if you have experience with making your own it won't take you long to come up with new adventures for them.
Fiasco (though you'll want to use different playsets than the book contains) is a very light but fun story-focused system. The main playsets are not at all kid appropriate, better ones for that age range are available online. It also has a stronger focus on imagination and storytelling versus mechanics (really, the mechanics only exist to set up the scenario and resolve a few things at two points in the play).
http://fiascoplaysets.com/genre:kid-friendly - I have no experience with any of these, so I can't comment on quality and actual kid-friendliness.
Dread is technically a horror game. But the mechanics are ludicrously simple. As the game master you talk to the kids and help them develop characters, and relationships between the characters. Then you put them in a situation. Whenever they do something challenging, they pull from a Jenga tower (7-8, this may be more difficult for them to do well, but there are other variations). If the tower falls, the character is out of play (in most Dread games this means dead, catatonic, severely wounded; with kids you may want to bring them back in but have the character unable to do much for a few rounds). Kids would probably have fun with it as the tension rises with the tower.
Both Dread and Fiasco can be used for any genre, fantasy, contemporary, horror, scifi. Just steal the mechanics and fit the experience to the kids interests.
Also check out https://rpggeek.com and https://reddit.com/r/rpg. Both are good sources for information like this.
The best system is really some combination of: A) what setting do you think they will enjoy? B) what would you most enjoy running for them? C) given the huge variety in systems, what style of gaming are you aiming for? ("Crunch heavy" versus storytelling-focused, simulationist versus abstractionist, etc)
Some other ideas to throw into the pot of interesting starter systems:
Mice & Mystics is a self-contained boardgame version of an RPG with cool miniatures and an interesting story. Compares to the Descent boardgame mentioned by someone else, but with a stranger setting. (Mice adventurers battling roaches and cats, etc.) (https://www.plaidhatgames.com/games/mice-and-mystics)
Do: Pilgrims of the Flying Temple can be a gentle introduction to shared storytelling (and encourage some writing skills work): https://www.evilhat.com/home/do-pilgrims-of-the-flying-templ...
Fiasco is a great rules-light storytelling game, that can be adapted to a younger audience (most of the playsets are relatively mature, given the basis in Coen Brothers cozy fiasco movies). (http://bullypulpitgames.com/games/fiasco/)
QAGS is a very simple system (single D20 for each player and a bag of candy for the GM): http://www.drivethrurpg.com/product/28315/QAGS-Second-Editio...
The Basic Roleplaying System is what it says on the tin (d100/2d10 percentile roles) and you can describe the system on a cocktail napkin almost. There's a ton of content already tuned for it, especially if you have interest in Lovecraftian settings.
I've been told that the Pip System of Third Eye Games was designed with young players in mind and built to scale up with player age/attention/interest. https://thirdeyegames.net/
Savage Worlds is the most "crunchy" system I would typically recommend, and there are a number of good settings. It's goal is "Fast. Fun. Furious.", and it has a pretty good ramp from very simple system rules to some more complex ones as Settings/Scenarios warrant them: https://www.peginc.com/product-category/savage-worlds/
A current infatuation of mine from a systems design standpoint (it is closest in spirit to an unfinished design of my own I've been meaning to finish) is CAPERS. It currently only has one setting (1920s "gangster" superheroes), but a unique, easy-to-learn system (press-your-luck playing card flipping): http://www.nerdburgergames.com/capers/