Being a family man and a DM/GM, I have not made special accommodations for my children by changing the game system used. I simply adjust the burden of the system mechanics I expect the child(ren) to shoulder. I have my children, as young as 5 describe to me the hero they wish to play. I then build the character for them based upon their input and my asking them pointed questions. The same method applies to game play. They describe their intended character actions and I, with help from other players, handle the game mechanics to make it happen while letting the kids handle the dice. My kids have picked things up readily and introduced novel ideas to the adventuring party (They have not bias as to how it "should" happen. All you need for this plan are patient and helpful mentors in your player group.
Specifically to give answer to your question about adventures and table-top games: Use any game system that you and your players are familiar with. Use adventures that are neither vulgar nor morally vague. I believe the players you have adventure alongside your young ones are equally as important as keeping the adventure in the right tone. Seek players with young families as well. Two generation games are solid fun and of equal value to adult only gaming. RPGs need not be rude and crude. I daresay most adventures mindfully GMed can be less offensive and more kid appropriate than Spongebob Squarepants and his ilk.
Running a child-friendly campaign is, at its base, simply an extension of good game mastering and leadership.
This past May we ran a revitalized campaign with my 3 younger children (11, 9, and 7) as participants. I had 2 other adults playing and I sat as the GM. Things went well. I would like to add that there may need be some work done early in the game to get the younger folks engaged early on. I lost their interest for some bit as we got the nuts and bolts introduction of both PCs and NPCs done. We (the adults) could have done better prep work to save the kids from enduring setup. Some adventuring woke them up. The battle that day they loved! My 11 y/o was happy to score the killing blow.