App Store on iOS keeps asking for another user's iTunes credentials
When an iPhone is restored from a backup, it gets as much data as is can locally. Any app data is stored in the backup, but the apps themselves either have to be synced from a local iTunes (preferred) or downloaded.
When an iPhone is synched with iTunes, by default all apps on the iPhone are copied to the iTunes library, assuming iTunes is registered with the same Apple ID as they were purchased with. I think.
iOS app files are registered with a specific Apple ID, and without this same authentication, there's no (easy, non-jailbreak) way to use the app with another Apple ID.


So, when you restored your friend's iPhone with your computer, rather than download all of the apps from the iTunes servers, it copied as many as it could from your computer. These apps are linked to your Apple ID, so in order for your friend to use them her iPhone would need to be registered with your Apple ID.

The solution: Your friend can back up her iPhone to iCloud or another computer (make sure, if so, that the offending apps are deleted from the iTunes library after synching) and restore it again. This will definitely keep her data from the other apps, and probably from yours as well. Alternatively, she can delete all of yours and re-download them from the App Store, which will definitely lose the data.