iPhone repeatedly asks me to sign in with old family Apple ID

Each application installed remembers the account that purchased it. The likely cause of this is one or more apps needing to update or to run and validate the purchase receipt (even free apps are purchased and get receipts).

• Once you delete those apps - you have to re-download apps from a device signed into the App Store account you do want to enter the password as opposed to the one that originally was involved in the purchase.

If you're not sure - you can make a solid backup (perhaps to iTunes as well as iCloud) and then delete apps you know you can get back from your purchase history - then focus on the remaining apps to figure out which you can discard entirely.

Now you're left with just the uncertain ones to troubleshoot. Once you know each - wipe and restore from your main backup to get everything back and then surgically remove the offenders.

Lots of work, but that's how I've solved this in the past. This is how it works on the Mac App, iTunes, iBooks and iOS stores...

• App Store asks for old Apple ID when installing updates