My wife has been watching All my Children for 27 years (including the 15 that I’ve known her), and couldn’t answer this question:
What’s with the name of the show? All who’s children?
Hmm good question…I used to watch it with my roommates and the guys across the hall in college. We were addicted! I would like to know the answer as well!
Per www.faqs.org:
Q: Why is it called “All My Children?”
A: Some people think the “My” refers to Kate Martin or Phoebe Tyler Wallingford. But what does the creator of the show have to say about it?..
Oprah asked Agnes Nixon that question when she had a bunch of them on a special commemorating the 20th Anniversary special back in January of 1990…Agnes Nixon replied:
"All My Children refers to the Brotherhood of Man (MY is God). You are All MY Children".
The Great and the Least,
The Rich and the Poor,
The Weak and the Strong,
In Sickness and in Health,
In Joy and Sorrow,
In Tragedy and Triumph,
You are All My Children
Agnes Nixon wrote this at the beginning of the first episode but unfortunately it never aired.
*
So now we know 
Actually, the poem appears in every episode of All My Children. If you watch closely, you can see it on the flyleaf of the photo album that flips open during the credits to reveal all of the actors/characters.
Well, there you go.
Thanks!