Can I renew my Non-Immigrant O visa based on marriage in Thailand or Australia?

First off you NEVER renew a visa ever. Visas run for a specific period of time then they expire.

You could apply for a yearly extension of stay based on marriage at the thai immigration office in your area by meeting the financial criteria of either 400K baht banked in a thai bank account in your name for 2 months prior to you applying for the extension OR by getting an affidavit of income notary letter from your consulate stating you make more than 40K baht a month.

You could also go to the thai consulate in Savannakhet Lao and get a year-long, multi-entry Non-O based on marriage without showing any financial proof at all. BUT with that visa, you'd have to exit/re-enter thailand every 90 days for the duration of the year.

You might also be able to get that year-long, mutli-entry Non-O back in Australia from a thai consulate, but you'd hafta see what their requirements are for issuing it.