Hi,
Just to clarify, did this person enter on a partner visa and have since split up with the sponsor and therefore, is remaining in Australia illegally? Does the Department of Immigration know about the change in family circumstances affecting visa conditions?
I do not see how the lack of visa status would prevent this person from initiating court action per se. Access to the court system is not only reserved for Australian citizens or those with visa rights. However, this person may risk being deported when the Department of Immigration finds out. Hence, I suggest that this person speak with the Department of immigration first and explain the situation. Perhaps they are able to re-enter Australia on another visa or for "tourism" purposes in order to initiate and complete the custody matter in Australia? Otherwise, this person is able to begin the action in their home country and seek to have that judgment enforced in Australia.
Another point to consider, this person's status in Australia and ability to remain or return to Australian may affect his or her custody claim. This is because if the child is here in Australia and has a family, friends, stable environment here, it may not be in the child's best interests to uproot the child off to another country. What are the chances of the child seeing the other parent or the rest of his or her family? Further, not disclosing information to the Department of Immigration may been seen as a dishonest act that may affect the person's character which will certainly be unfavourable to the claim. These are all factors the court will take into account.