I've now had a chance to try to get my daughter's Apple ID changed, but I've not managed to solve this yet.
You mention that free @icloud.com addresses are assigned but is that not just for those that already have an @mac.com one? My daughter doesn't have one since her main PC is a Windows one and when she first signed up for an Apple ID she used another email address, the one that is now defunct.
What we've done so far is to go to the appleid.apple.com and clicked on Manage your Apple ID and signed in using her old defunct email address. On the Manage your Name, ID ..... page there was no Alternate Email Addresses shown so no @icloud.com address was there. She clicked on Add Email Address and entered an @icloud.com email address that she thought might be acceptable. It was and she got back an email to that new address confirming that it had been set up. However it just appeared under Alternative Email Adderesses and not as an Alternative Apple ID, so she couldn't delete the old defunct address.
She went back to her iPad and found that we could have set up an Apple ID there by using Settings/Mail, Contacts, Calendars/Add Account and choosing iCloud from the list and then clicking Get a Free Apple ID. She did try this but found of course that she couldn't use the same icloud email address that she'd just set up, and appararently wasn't able to set up another one.
I'm seeing her again tomorrow and we're going to have another attempt at setting up a new Apple ID using her iPad, and if this is accepted hope that we'll be able to change her Apple ID on the appleid.apple.com website. If you have any comments on what we've done so far that would be very welcome.