Mineral Badge - Pokemon Soul Silver
Olivine City / Gym
Go to the top of the Lighthouse (you can take the elevator this time) again and give the SecretPotion to Ampharos. Jasmine will leave and head back to the Gym. In a strange twist all the Trainers in the Gym as so glad you helped Ampharos that you don't need to fight anyone but Jasmine straight away.
lvl 30 Magnemite, lvl 35 Steelix, and lvl 30 Magnemite vs. lvl 36 Typhlosion
Pretty simple for where you are level wise. Make sure you can do some good damage to a lvl 35. Once you beat her, she will give you the Mineral Badge.
Mineral Badge - Badge Battle