The lock screen is the screen you look at when the phone is locked. Some (all ?) handsets allow you to choose the picture so you create a picture with the number of the person you want the person with your phone, to pester, and use that.
I've figured it out. On iphones theres a health app. you can fill in as much or as little medical info as you wish into the app, including emergency numbers and have them appear behind the emergency tap on the lock screen.