Q: Handed down iPhone 4s

In what update? If it's in an update to an app, it's because you somehow have an app on your phone that's tied to the previous owner's iTunes Account. If it's asking for the password to their Apple ID, it's because they never properly removed it from their account.

This article outlines what should have been done before the iPhone was turned over to you:

What to do before selling or giving away your iPhone, iPad, or iPod touch - Apple Support

Since iOS 7, antitheft measures were incorporated to reduce the great amount of robberies and assaults to get the profitable iPhones. So now if the device is password protected, it becomes useless to the thief. With the downside that if the device is not adequately released from the custody of the former owner, it too becomes useless to the unsuspecting new one. See here:

What to do before selling or giving away your iPhone, iPad, or iPod touch - Apple Support

Turn off Find My iPhone Activation Lock - Apple Support