The adult's $6,000 tax-free threshold per income year applies to child.
Your tax-free threshold may be less than $6,000 in a financial year if you are not a resident of Australia for tax purposes in the whole income year.

However, Children are not exempt from quoting a TFN when earning interest from saving accounts.

Children's bank accounts
Children are entitled to a threshold of $420 per financial year on interest. If a child earns more than $420 interest from his saving account during an income year, he needs to quote a FTN. Otherwise the bank or the investment body will withhold PAYG at 46.5% for the total interest earned (not just the amount above $420). In this instance, he needs to get a TFN first and lodge a tax return to claim the amounts back from ATO.

At what age can a child get a tax file number?
There is no minimum age. A child can get a TFN at any age.

Source from Australian Taxation Office

Edited by user Saturday, 30 October 2010 2:39:47 PM(UTC)  | Reason: Not specified