Are you attempting to flash a stock ROM, or a custom ROM?
Firstly, have you determined what version you have?
gt-i8190, gt-i8190L, or gt-i8190N?
if a custom rom you'll have to flash TWRP with Odin which will allow you to load custom roms from a ZIP stored on the microSD card.
My wife's s3 mini was flashed with a custom rom a long long time ago, not sure if I still have the files on my NAS at home, but i can take a look. Though XDA has the majority of the info, but I imagine many of the links are broken now....