After 8 months, she says she want to throw away the hamster AGAIN? Best answer for the best answer!?

Your mom is just OCD about having animals in her house, there is nothing you can say to change that. My mom used to go through my room when i was little and just randomly throw stuff away because she didnt like it for one reason or another. Little rodents just bug her and that wont change not matter how quiet or clean you make the cage. Theres nothing you can do except just keep arguing until one day she'll hopefully just give up. If shes willing to ruin a relationship between you two over hamsters then you should not let her win