CyanogenMod community has released CM10.1 for the Galaxy S3 Mini users. This version that is still in development on xda is built for the 8GB GSM version of the Mini (GT i8190) running I8190XXAMA2 baseband. Make sure to make a complete backup of your phone before you proceed as this ROM is still in beta and you may loose information on your phone if you do not do a backup.
You can download the ROM from here. For those who do not know, CyanogenMod is a community built firmware that is not supported by OEMs and manufactures. On the bright side, it is designed to improve the overall performance.