Kungfu_Teddybear said:
Try Demon's Souls and Dark Souls.
I tried to introduce my mother to Dark Souls... I think she liked it! And she only died once! Mind you, this was the Undead Asylum first level/tutorial.
My suggestions? Try
Adventure Quest or
Dragonfable, they're free to play, fairly simple with more depth if you get into it, and pretty much aimed at kids. You can kill most of the early enemies by just spamming the 'Attack' button.
A fighting game is also a good choice, button mashing can get you through weaker AI no trouble, but mastering mechanics can take a while.
Finally, platformers. Try any of the Mario platformers, Spyro the Dragon or possibly the Fancy Pants Adventures series, which you can play free online.
EDIT: World of Goo, anyone familiar with physics and gravity can play it. Also, I've heard
Pokemon is good, though I'm afraid I've never played a Pokemon game before.
My current plan is to introduce my parents to Resident Evil 4, as provided by the Playstation Store. They're both Doom alumni, and Resi4 doesn't use the dual analogue stick control system.