I am very firm about this issue. I have an Xbox, but I think most of the games that we like are available on PS2 as well. Our favorites are:
Tak 2. I haven't played the original Tak but from all reviews that I found, Tak 2 is a huge step up anyway. We love this one. No violence at all.
Morrowind. This game does have fighting in it, but it is entirely optional. We play it in a very non-competitive manner, and just leave if someone comes looking for a fight. We still enjoy the game a lot. It has won awards.
Urbz: Sims In The City. This one may not be really suitable for littles, it has some adult themes, but for older kids I don't mind the kissing scenes or what have you.
Sonic. We just love Sonic the Hedgehog

No violence whatsoever.
We also have a soccer game and a tennis game. Neither of those are violent at all.
Harry Potter. There are several of these.
And last but not least, we have a pinball game that we like, as well as some racing cars or spacecraft type games. There are violent ones out there, but the ones that we have are totally fine.
When I am looking to buy a new game, I just go to Blockbuster or what have you and browse. You can get a pretty good idea by reading the jacket carefully and looking at the screenshots.