Hello and thank you for using Just Answer. My name is XXXXX XXXXX I will be happy to assist you.
I am very sorry to hear about the situation you are in. I am going to try and explain some networking information to you so you understand a bit better.
First, you do not change your public IP in your router. Your public IP is issued from your ISP to your router. Your router issues internal IP addresses to your computers but the IP address that is showing from your location IS issued by your ISP.
There are a couple of ways for you to change this yourself, you could use a number of different proxy websites which would allow you to connect from a different IP address.
There are also software solutions as well, here is a 2 minute tutorial on changing your public IP in order to get around a ban, http://www.youtube.com/watch?v=84WSZLP9WBI
All of that said, how did this happen in the first place? You are asking if it is technically possible for someone to appear to be located at your IP address, the answer would be yes. All the person would have to do is connect to your wireless network in your home. Whatever they did online would appear to come from your public IP.
I guess the question would then be, does your daughter have any friends that are neighbors perhaps and would maybe want to be mean to her? Hacking like this is not a random occurrence. Hacking is also not super easy and takes technical know how. So if someone is doing this to her I would assume they know her and are doing it on purpose. I can't see a random stranger taking the time to hack your local network to get a girl banned from a website.
Let me know if you think there is any chance someone close to her would maybe be mad and do something like this?
Also please feel free to ask me follow up networking questions if there is anything you do not understand, I am here to help.