In the Roman Catholic tradition, this problem is solved with godparents. I'm not Catholic (and I don't have kids yet), but the model seems to make sense. Forgive me for stepping way beyond the bounds of "internet stranger's business" but if your close friends' lives are so full that they would not care for your children if you became incapacitated, you need better friends.
I come from the RC tradition as well, though we didn't baptize our children.
I can think of exactly two sets of people in our diverse set of friends (and they are good friends) whom we might ask for this kind of consideration, but really, even both of them are problematic for reasons that have nothing to do with the quality of our friendship.
I strongly disagree with that view. I don't have children of my own and while I would open the door at 3 am to take friends in (or heck a neighbor) while we get things sorted out taking on children is a major life disrupting experience.
So yes saying no would be selfish, assuming that any of your friends would have their lives completely disrupted just because you have children seems quite a bit more selfish.
The point is that throughout history, raising children are considered the point of life, not trophies that are otherwise a distraction from vacations and toys.
Well, no. Traditionally care for those kids goes to the extended family, not the friends. Sneering at friends who won't take in children is not something that you can support in terms of an argument of tradition.
I don't think it would solve the problem you are responding to, which is that the person doesn't know anyone they feel are capable of taking care of their children. Having a formal title for the role isn't going to help them find someone for the role.
