Android 10, if and when you can get it, makes setting Emergency Contact Info a breeze — no need to dig deep into Settings anymore.
Commonly referred to as ICE info, In-Case of Emergency, this info is easily set by holding your Power button and choosing the ✳ Emergency button at the bottom:
This opens the emergency dialer, which is restricted to only call emergency contact numbers. The point being if someone was to use your phone while you were incapacitated, they could contact emergency services or people you’ve designated as emergency contacts on your phone.
To set those emergency contacts, choose the Emergency Information button at the top of the dialer:
I’ve cropped my information out of view, although rest assured that if you were to find me unconscious, you’d be able to see the details without unlocking my phone.
In addition to the emergency contact info, I’ve also set a lock screen message on my phone with alternate contact info (email, twitter), so that if my phone was to become lost, a good Samaritan would be able to contact me and let me know.