Yesterday I heard a disturbing story of an acquaintance who suffered from a brain aneurism and was found stumbling around a nearby campus with no recollection of who she was or how she got there and why she was so confused. Luckily on her phone there was a screen saver message with a simple phrase: ICE: and a phone number. The police and paramedics recognized this phrase as "in case of emergency" and called the number to begin getting family involved.
This simple precaution can also help in a case as pedantic as if you loose your phone or drop it off the chair lift.
With an iPhone:
Step 1: open the notepad app, create a new note. Skip a few blank lines and write your emergency contact message.
Step 2: press the power key and within a second, simultaneously press the home key. This makes a snapshot of your screen.
Step 3: go to Settings, then select Wallpaper and scroll through available images until the snapshot if your emergency message appears.
Step 4: pick that image, then click on SET, and select "lock Screen". This will make the image of your message the screen saver when your phone is locked.
Now, whenever you awake your phone and swipe the slider, behind the keypad and entry for your unlock code will be your emergency message. Even if the person finding your phone does not know your unlock code, they will have some information to help unravel the mystery of the orphaned phone or it's confused owner.
Similar operations should apply for an android or other phone. If you don't lock your phone, make an entry in the phone/contact directory for "In case of emergency" and one for home with a number and a note.