Hi Newman,
There's two types of activation involved:
There's activation with the cellular service. For that you should contact your cellular provider. It's been a while since I did that but if I remember correctly I needed an Apple ID not because the provider required it, but because I had to get into the phone to do all the setup and getting service flowed out of that. I'm not sure it's 100% necessary but it might make life a lot easier to get an ID set up on the phone first.
There's activation of Apple services on the phone. For that she will need an Apple ID and ideally she should use the one she had before. Doing that will optimize her phone being repopulated with her previous settings and downloads. She can activate Apple features with a new ID but then it will be a brand new account and it isn't easy to transfer things from an old account should she get that working again. So in my opinion it would be best if she can find her old ID and use that account.
Typically an Apple ID is an email address you have used.
Click on this link to read what to do "If you forgot your Apple ID" --> If you forgot your Apple ID email address or phone number - Apple Support