If you like keeping your console, I suggest a PS3, or a Wii. Four letters. RROD. The consoles' failure rate is staggering!
I like "keeping my console", and I love my 360 just fine. I got the RROD, but I still have an Xbox. If it fails, you get a new one for free. Two weeks without an Xbox, then bam, a fresh one at your doorstep. Hey, if I knew with certainty it was going to happen, and when, I'd still get it. What happened after the RROD was absolutely no different than if I just abstained from playing for a couple weeks (plus getting a free month of LIVE

). If you compare the RROD experience to all the rest of the time I've had a 360, it could hardly be considered even a minor inconvenience.
I've had my Wii since 2007, and use it regularly. No problems whatsoever.
If you got an Xbox instead at that time, you could quite possibly be saying the same about it. I play my Xbox regularly, and when I had had my Xbox for as long as you had your Wii, it had always worked perfectly.
I played Zelda Twilight Princess for almost 6 hours in one sitting, and had no issues.
You can play the 360 for 6 hours straight with no problem, too.
I plan on getting a PS3 next. That is definately the safest next-gen system to get. XBox has great exclusves, but I don't want to pay $249-$299 on a console that might not work when I want to play after work.
Yes, PS3 is safer. But the Xbox is great, too, even with the RROD problem. You said it yourself, it has great games. If you truly think the games are great, and you would love to play them, get an Xbox. If the RROD happens, you have to go a max of 2 weeks without your Xbox (and if that ruins your life, you need to go outside once in a while). They replace it for free. I love Halo 3, and I think it would have been so stupid of me to not get an Xbox (and thus not play Halo 3) just because of the RROD thing. The minor inconvenience the RROD caused me is nothing compared to all the good I've gotten out of my Xbox. If the RROD is the only thing keeping you from buying a 360, then the 360 must not be so great to begin with.
However, in regards to the original poster, the Wii is the one to get if your child will be the one playing the most. Get the Wii.