[Marticus]

I can think of an iPhone immediately (cringe) and how you can make an emergency call when the phone is locked.

Have it just display something like "Emergency contact" who you can dial without displaying any more information on the person, not even the phone number when it dials.

Having this person know personal information about you (blood type, etc) would then still be relatively secure versus openly available, and the person would have little risk of getting prank called if your phone is stolen outside of an emergency situation.