Can children fly with ESTA?

Children of all ages, including infants, are required to have a valid ESTA when travelling to the United States, if citizens of a Visa Waiver Country, in the same way as adults are.
Children flying with ESTA may not fly under either of their parents’ passports. The child must have an unexpired passport which must be machine readable and possess a digital photograph of the bearer integrated into the biographical data page. These are commonly known as either biometric or electronic passports. The passport must have a minimum of one blank page for the stamp. If the passport is to expire in less than six months, then the authorisation obtained through ESTA, which is normally 2 years, will only last until the passport expires.
A person under 18 years of age may not complete the ESTA on their own. Any such application completed without assistance from an adult will be automatically refused. When applying for a child, this can be done separately, but is normally carried out with yourself through a group application. If you are filling out the form for a minor, please make sure to tick the relevant box applicable to Third Parties with respect to the Waiver of Rights, as is detailed further in Can a third party fill out ESTA for me?.
An adult who is applying for a child must state their relationship with the child during the application process.
The fee for applying for a child is the same as the fee of an adult application. The same is true for the validity period of the authorisation, and the response time from the authorities.

Where the child has a different name from the adults they are travelling with?

Where a child is travelling with adults who are not their parents, such as grandparents, a divorced parent, or family friends, then the adults will need to provide other official documents for the child to make sure that the child is allowed to travel with them and that they can enter the United States with them easily.
Where a child is travelling alone without their parents, then a letter of authorisation to leave the country signed by the child’s parents or legal guardians of the child together with copies of their passports or identity cards are required.
This is also the case where a child is travelling alone with one of their parents, particularly if their family name is not the same as that of the parent with whom they are travelling. In such case, it is advisable to carry a copy of the other parent’s passport or identity card along with a letter of authorisation signed by them authorising the child leaving the country.
In general, it is best to carry copies of all documents verifying your ties to any child that may be accompanying you.