Q: New iPhone how do I back up without effecting my itunes

If you back up her phone in iTunes on your computer you will have a backup called Mums Phone (or something like that) it will contain what it says, nothing more.

Do you mean a sync when you say a backup. Do you know how to perform a backup?

If you do what Csound1 suggested, then none of the media and third-party applications stored on the old device will be transferred to the new one, except for media recorded using the device. You should instead create a new OS X user account and then transfer the data using it. Additionally, you should disassociate your computer from your mum's iTunes Store account after performing the transfer.

When you want to perform the transfer, log in to your mum's user account, connect the old iPhone to the computer, and then choose to synchronise the device with the computer and transfer your iTunes Store purchases. iTunes will automatically create a backup of the device's data when you synchronise it. Once you've done this, follow the instructions in the second article I referenced in my previous post.