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For an 8 year old, I have no idea how they can figure out all those buttons so easy for video games unless they grew up with them.
I was 11 when I got a SNES, and I did fine with it, and I didn't have anything else before.
I recommend an NES or an Atari 2600. You can get tons and tons of games, probably get them cheap, you may need a little work done, but you can get good ones at flea markets for like $10 and games for a dollar each. Atari has Asteroids and Pac Man, all that good stuff, and retro is always better...
There are little devices that you can find just about anywhere that have old Atari, Activision, or Sega games on them. They are just the controller that you hook up to the TV.
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I appreciate your concern...however she is already quite involved in extra-curricular activities. She takes Dance, Piano lessons, and a church related activity every week. Also swim lessons when the fall comes around. She has to READ aloud to either Mom or myself for at least 30 minutes every night.She isn't allowed to watch TV at all during the week. And when she does there are only 3 channels she is allowed to watch. She loves to play games on the computer and when she does....then I can't post on UM. Actually she is a very well rounded gal and the video game console will not 'become her life'. But I wanted to make sure that what I got her would be Kid Friendly.
Your kid likes what I call "short games". Those are arcade kinda things; really simple and are easy to start playing and easy to stop.
Your kid seems to have a lot to do already, so they don't have much time to devote to console games. (Which tend to be longer and more complicated than internet/arcade type games.)

These have ten games: Centepide, Asteroids, Missle Command, Circus Atari, Yar's Revenge, Breakout, Gravitar, RealSports Volleyball, Pong and Aventure.
Otherwise, a Nintendo Ds would be good cause most of the games for it are those short and simple ones. And then there's Nintendogs!

Puppy!!
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Well just viewing some of the pictures online I thought maybe they had their own screen..like a laptop kind of thing...I told you I didn't know anything about them...
That's alright, some third-party companies (MadCatz, Pelican) make screens that you can attach to your console so you can take it with you on vacations and stuff.
But I don't have one 'cause they cost a couple of hundred, and I don't leave my house for long much. (PS: The Nintendo DS has two screens; the top one is just a normal one, and the bottom one is a touch screen.)
I hope that helps you out some.