Others have already given the answer:
You purchase a life insurance policy to replace income. The vast majority of children do not bring home an income.
For example, a father who is the sole support of his family MUST have life insurance; if his income was lost, his family would be devestated financially. The mother in this traditional-family model should also be insured because if she should die, the father would be forced to pay (quite a bit) for services she is now providing: child care, transporatation, etc. But consider the children in this situation . . . if one of them were to die, the family would not be worse off financially. They would not need to replace income.
It is wise to have a SMALL burial policy. Through my work, I can buy a $10,000 per child policy for $1 per family per year.
That's a throw-away amount of money. Before you start spending real money, calculate up what you'd have if you invested this small amount each month. What's more likely? That your child would appreciate that money when he's just getting out on his own, or him dying young? Also, they promise that your child will be able to continue this insurance as an adult . . . but do they say at what price? It won't be that low price you're paying now. If the child turns out to have a condition that makes him difficult to insure, they can simply price the policy out of a middle-class person's reach.
Bringing up a similar subject, you're much more likely to become disabled than to die while your children are small. In addition to life insurance, adults should have disability insurance. Going back to the above traditional family example, the nightmare situation isn't the father's untimely death -- the financial nightmare situation is that the father becomes disabled, runs up huge medical bills, needs extensive medical care, AND is not bringing home a paycheck OR collecting on his life insurance. Mom, now caring for both children and an ill spouse, may be unable to go get a job. Without disability insurance, this family is screwed.
All this sounds cold-hearted when you discuss it, but it's important.