You said it flew across the room. Is it not clipped? As much as I don't like clipping and think birds should be allowed to fly, I don't think an inexperienced owner or 7 year old can really deal with a flighted bird.
It may have been better to reach into the box and grab the bird rather than have it fly out and be scared while you chase it around trying to get it.
What kind of food did you leave in its cage for it? It may be a good idea to leave cockatiel seed mix, millet spray, and moistened cockatiel biscuits until it settles down. If it's a baby, definitely leave cockatiel biscuits and if it's a bit older, under the stressful changes it may still be good to leave those initially.
Be sure to keep us posted what happens. I'm sure everyone is curious how your niece reacts and then gets along with the bird. If everything works out, perhaps it will change our views about ownership so it will be interesting to see what happens. If you get any videos of the cockatiel or your niece with it, feel free to share. Just remember to tell your niece it's not like getting a toy for christmas. You can't play with it all day and overwhelm it at first and then get bored of it later on.