Brand new iPhone prompts me to enter a password to an unfamiliar apple ID ??

It's because you downloaded pirated media that had been purchased from Apple by the person with the Apple ID you saw and then distributed illegally. Their ID is still stored in the files, so you're being asked to confirm that you in fact own the files.

To verify this, check all the iTunes media you're syncing to your phone (most likely music), right-click and 'get-info' for all your music and if it was purchased by that person after a certain date, you'll see their ID listed in the 'File' tab as the 'Purchaser.' Media purchased before Apple started storing this info with the file won't have that person's ID stored in the info, but it will still cause the conflict.

When you restore a backup it downloads purchased apps and other media that were on the device that was backed up. Before it will do this, it requires the password for any Apple IDs used to purchase this media. It looks like you may have had purchases from these other IDs on the device that was backed up.

Did anyone ever sync their device with your iTunes library? If so, it may have transferred media to your library that was purchased with these IDs, and they were subsequently synced to your old phone and stored in the backup. If you keep skipping the prompt for the IDs it should eventually stop prompting you. The consequence of this is that any media purchased with these IDs would not be restored to your new phone.