Yes. An eVisitor is linked to one passport, so every member of the family needs their own — including a baby travelling on a parent's lap.
How to apply for a child
The application is made in the child's own name, using the child's own passport. A parent or guardian can complete it on their behalf. The visa is free for children as well; there is no separate fee.
Check the passport first
Children's passports are often issued for shorter periods than adults'. A passport that expires during the 12-month validity is a common cause of trouble later, because the visa is tied to the passport it was granted against.
If a child travels with one parent or with someone else
Australia may ask about consent arrangements at the border. Carrying a letter of consent from the non-travelling parent, and a copy of any custody documentation, is not a visa requirement but it removes a predictable delay.
Source: Department of Home Affairs — eVisitor (subclass 651)
More on the eVisitor
What is the eVisitor
The eVisitor is a full visitor visa granted electronically to holders of eligible European passports. What it covers and how it differs from the ETA.
How to apply
Applications are made free of charge through ImmiAccount on the Department of Home Affairs website. The order of the steps and the mistakes that cause delays.
Cost
The Australian government charges AUD $0 for the eVisitor: no visa application charge and no service fee. The ETA for other passports costs AUD $20.
What you need
A valid European passport, an email address and about fifteen minutes. No photo, no payment, no documents to upload for most applicants.
Processing time
Most eVisitor applications are decided automatically within a day. What causes an application to take weeks instead.
Validity
An eVisitor lasts 12 months from the date of grant, or until your passport expires, and allows as many entries as you like within that period.