A child may be eligible for a variety of Social Security benefits including survivor benefits, disability benefits, and even "retirement" benefits.
For example, your unmarried children who are under 18 (up to age 19 if attending elementary or secondary school full time) can be eligible to receive Social Security benefits when you die.
Children from birth up to age 18 may qualify for Supplemental Security Income (SSI) benefits. They must meet Social Security's definition of disability for children and the parents and child must have low income and few assets.
Additionally, if a parent receives Social Security retirement benefits, an unmarried child under age 18 may receive payments based on the retiree's work record.
Finally, if the surviving spouse or the spouse of a disabled worker is caring for a child under age 16, then the care-taking spouse may be eligible for Social Security payments.
In general, payments for children are made to a "representative payee" (typically a parent) who must use the funds for the benefit of the child.