I want to give my niece my old iPhone 3GS so she can use it as an iPod touch. I don't want it connected to my iTunes.
Should I buy a new SIM card for it or should I leave my old one in?
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A new SIM card is only necessary if you want to still use it as a phone. You don't have to change the SIM card (but do leave the old one in place) if all it's going to be used for is an 'ipod touch' type experience. Your cellular provider will have turned off that SIM when you activated your newer iPhone.
Take your old 3GS and go to 'Settings > General > Reset > Erase All Content and Settings.
This effectively makes it a new phone. No apps will be present other than the default apps that are on all new iPhones. There will be no record of your appleID on the phone.
Now if your niece wants to buy apps from the app store she'll need to set-up and use her own apple ID.
See also this thread: https://discussions.apple.com/thread/3399341