The HABA publishing series is always a hit. Rhino Hero working for kids and adults alike.
First Orchard teaches the basics of game mechanics to younger kids.
Set Junior simplifies Set for kiddos.
Ice Cool is a great dexterity game.
The Magic Labyrinth (2009) involves mental mapping and memorization skills.
Looney Quest is a great drawing game as you try to collect coins and avoid the bad stuff.
Not to mention all the Junior games of heavy classics like Ticket to Ride: First Journey, My First Stone Age, etc.
Obviously the downside is they likely will outgrow these simplified versions. So the value of hacking the games you love as an adult is definitely a big plus and lets you save money and work them towards great board games. But on the other hand, why strip out all the pieces when you can play something designed for that age demographic?
There is also the fact that some games won't even have reduced sets (Guillotine for eg), or will have it but it will be impossibly hard to get where I live. I can easily get a copy of TTR in India, but finding a copy of TTR:1st Journey is much tougher.
All the standard rules, no-one can modify them, runs the auctions for you, plays very fast. D-pad and four buttons (counting select and start) so even young kids (or oldsters) with no gaming experience can pick it up fast. I wouldn't ordinarily recommend a video game as a complete replacement for the physical version of a board game, but in the case of Monopoly, yes, definitely, find the simplest video game version you can and use that.
Monopoly is not a well-designed game to play with people but, surprisingly, amazing and interesting to play against an 8-bit computer.
Other games like Pigasus and Snowman Dice (both by Brain Games I think) have been a hit too.
When we play with younger kids we try really hard to emphasize fun and downplay stringent rule following, for better or for worse.
Some games, like snowman dice, are fun just for the sake of it. It's fun to roll the dice and push them around.
I'd say Catan Junior is fine for a 3 year old, but make sure to spend the first few times just familiarizing the rules (i.e. making the connection between getting a resource and spending it by matching up the resources on your resource card).
Ticket to Ride Junior is a little bit harder, but a 3 year old should do ok with help. Again, if you really "take it easy" about the rules a very young child can have a lot of fun with these if for nothing other than the colors and manipulatives/meeples alone.
Magic Maze Kids we did starting at 2.5-3ish and it was fine.
Pigasus, on the other hand, a 2 year old can do and really enjoy. That game is absolutely amazing because it's one of those things that a small child can do better than some adults (like... for real!). Also the sound the pig makes when you squeeze it is incredibly funny to most kids and very rewarding. It's a fun one.
OTOH, I typically teach adults how to play Race for the Galaxy by starting out with a few hands of San Juan... It's a very similar, but much much simpler game.
Another one that nobody believes me about (because it sounds like such a preposterous cash grab) is Super Mario Brothers Monopoly. But the changed some of the fundamentally boring aspects of the game to make it really quick to play and introduced the concept of "hero" characters with unique abilities to it.