Post-Setup (User) runs on first logon but you won't have elevated rights. When you're mapping network drives, it requires admin.
While you can try PS script self-elevation, you run into the UAC dialog which users can accidentally dismiss.
To save time, I will tell you how to write your script so you don't waste everyone's attention.
Check the PC's primary user in PowerShell (which family member), then inside your script automatically map their personal drive locally. Run this script during normal Post-Setup (Machine). No wasting time on admin rights. Unless the rest of your family knows how to code, a simple hashed password credential will prevent them from reading passwords.