iPhone 5S stuck in iCloud Restore

If may be the the old email she was prompted to provide the password for is an old Apple ID that was used to purchase one of the apps or other purchased media on the phone that was backed up. Any purchased media associated with that old ID would not be restored if you cannot provide the proper password. If you believe you have everying restored that you need I suppose you could go ahead and stop the restore process. Otherwise, you may want to take steps to reset the password for the old ID so you could provide the password when prompted (http://support.apple.com/kb/HT5787).

I had the same issue "This iPhone is currently being restored and will automatically back up when it is done." I reset and I downloaed the software update again. Seems to be loading and fixed...

Make sure that you have the latest iOS version installed before you start the restore. In my case the replacement unit I got had iOS 7.0.3 installed and the restore would get stuck at some point. I was able to install the 7.0.4 update while the restore was stuck using my Mac and iTunes. After updating the system, the iPhone restartet and continued with the restore. I hope it will complete the process eventually. It is still running at the moment...

See "iCloud: Restore your iOS device from iCloud":

You must update your device to the latest version of iOS to make sure you can restore from a recent backup of another device, such as a lost or broken device.

Good luck to you!

Next time I will make sure to have a manual backup of my device on my Mac before a planned replacement. Shoud be significantly faster than iCloud.

Here is my experience.

I restored my phone last night. I performed the "erase all content and settings" without upgrading first to ios7. I restored from iCloud. Everything went fine, but after a couple of hours I noticed the "This iPhone is currently being restored and will automatically back up when it is done" message in my settings. I checked all my apps and folders and music in iTunes and there was nothing that was still downloading. My Camera Roll was not set to back up to iCloud so it wasn't my pictures that was keeping the restore from finishing...or so I thought. When I saw that my Camera Roll was not set to back up and that I had lost all my pictures, I started going through old iMessages to see what pictures I could still save back to my Camera Roll. I had a couple of iMessages with family members that went back several months and contained many pictures. As I kept going back further in the message I noticed that when I clicked on a picture to save it to the Camera Roll it would just be a black screen that said "Restoring". That's when I realized that the restore was going back through all of my iMessages and restoring any pictures that were a part of the message. I let it keep working overnight and in the morning it was back to normal. And the good thing is that most of my pictures that were lost from the Camera Roll were still in those iMessages.

So that is just one more place to look if your restore appears to be "stuck."