If you are interested in staying in Australia beyond the 3 months granted by the ETA, you will need to make a separate application.
In order to stay beyond 3 months, you will need to apply for a Tourist Visa (subclass 676). The visa is for people who want to visit Australia or for other short-term non-work purposes including study for less than three months.
You can apply for this visa if you are in Australia and want to extend your stay as a visitor. The visa will allow the holder to stay in Australia from up to 12 months, or a shorter period, depending on the purposes of the visit and personal circumstances. The holder can also study on this visa for up to 3 months.
If you are in Australia and want to extend your stay for tourism purposes, you must apply for a new visa at least two weeks before your current visa expires.
A Tourist visa may be granted if you:
have complied with the conditions
of your previous visa
are not extending your stay to start, continue or complete a course of study
do not have a 'No further stay' condition on your visa
You should note that a further visa, if granted, will cease any visa or ETA currently held and the entitlements attached to that visa or ETA.
If a decision has not been made on your application before your original visa expires, you will be able to remain lawfully in Australia until the application has been finalised.
If you are refused a further visa, you must leave Australia before the expiry of your authorised stay period.
You must comply with the following visa conditions:
you must not work while in Australia
you must not study for more than three months while in Australia
if you intend to study for longer than four weeks, you may be required to pass a health examination before starting your course. Some countries are exempt from this examination
Your visa may be granted with a 'No further stay' condition. If this is the case, you will not be granted any other visa (except for a Protection visa) while you remain in Australia. This condition may be waived in exceptional circumstances