You'll need an invitation letter from your mother for a place to stay in Australia. And another from you saying you will provide financial support for the trip and ensure she obeys the visa conditions. And any other information you can give about her circumstances that would ensure she leaves, job, cars, property and family in Thailand. Good luck.
Do a border run and get stamped in for another year on your O-A You get stamped in for 1 additional year on each entry within the visa validity period. But you'll need a reenry permit if you travel outside Thailand during the second year.