iPhone X won't accept new Apple ID

You first need to fully “wipe” yourself out of that phone as follows:

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support

Then you can setup the device as hers.

The phone is not “ready” until you’re greeted with the script “Hello” and language selection screen at power-up.

HOWEVER …

… if her lost phone was running an iOS 17 or later, she’ll not be able to restore a backup of her presumably newer lost phone to the old X (which can only run up to iOS 16)

From any browser.......what happens if you try to log on to her Apple Account at iCloud.com and enter her Apple ID and Password?

If you can log onto the account that way, you know that the Password is correct, so trying to change the Password is not required and will just complicate things.

I really think you are going to need to set up an appointment at the Apple Store / Genius Bar to take in both the iPhone and laptop to see if there might be an undocumented workaround for a situation like this.

When you say that you "reset" the old iPhone, did you follow all of the steps in this Apple Support document?

What to do before you sell, give away, or trade in your ...Apple SupportOfficial Apple Support › en-us

I've never been asked to wait for confirmation via phone when getting an old iPhone ready for sale or trade-in

Yes.

We get the hello message and a start to set up as a new phone but the log on to Apple ID fails and trying to reset the password (which may mean she has trouble logging into her laptop,) asks for her phone number, which she has not been able to activate so couldn't see any link or access messages.

So it is a circular problem.

Her old phone was Android.

We get the hello message and a start to set up as a new phone but the log on to Apple ID fails

Why? Wrong Password? What is the message that you see when you try to log in with her Apple ID (now known as Apple Account)?

It doesn't seem to recognise the password, carefully copied from the Password Application on her laptop.

We tried to reset the password but the only question was the phone number, which we can not access on the phone yet.

we also tried to open a new Apple ID but her only email was "taken" of course, so we couldn't go any further.

going through a long process she has managed to change her AppleID password and log in.

Getting the phone SIM working has still not worked after hours "discussing" with bots.

She is 74 and would much prefer never to have a mobile phone but few things work any more without, even though they aren't much use with.

thanks for any help, the whole thing is exasperating and she still has no phone 2 weeks after hers was stolen.

Getting the eSIM / SIM on the phone working with the carrier will require contacting the carrier.

Hard to do when the phone is not working, so another phone will be required to reach the carrier.....or....a visit to the carrier's store would work as well, if that is an option.

Unfortunately, losing an iPhone is a very painful and frustrating experience.

Sorry she will try again tomorrow, the phone is off now...

Unfortunately, iPhone X was the final non-Chinese model to NOT have an eSIM capability …

… so a trip to the carrier store is even more likely.