Getting an old parent's visa is an almost impossible task. They tend to kick you out once you are very old and getting unwell, as we read from newspaper.
I understand your will, OP, to assess this possibility. Being a single mum does;t mean you have to suppress career dreams or moves. You should however try to make it worth for your son as well and aim at the "Australia" you have in mind when you think about it, so beach, a house with a pool, going to surf after school, going bare feet everywhere, so for this reason I would stick to Eastern suburbs which means Coggee, Maroubra, Bronte (avoid Bondi) or on the other side of the Harbour, Manly, Freshwater. Anything further up would make your commute harder.
The Lower north shore , the inner suburbs or the west, is maybe not the Australia we get from movies, series, documentaries , that would make it worth it.
Seriously consider Manly. Easy to reach the CBD by ferry and its popularity would mean you will easily find Aupairs. Yes it is busy and crowded but slightly move away from the Corso and not so much any more. Manly is also flat and everyone moves around in bicycle, great library, nice to go for a walk after work before dinner with your son. There are lots of expats, so many who share your experience and might be more prone to help in case of emergency.
Manly has a great new pool, plenty of parks, oval and is very friendly.
Aim at a beautiful place, both for you and your son.