I have tried probably about 20+ roms on the S3. I will mention 3 roms that I highly recommend. Starting with my favorite, I would easily say
Eclipse!!! Eclipse has
AOSP dark theming and it has the firmware built on
TouchWiz, so you can keep all the Samsung functionality. It looks awesome and has almost every Google app inverted! Lots of options here too
Second one I recommend is
Paranoid Android. It is a solid AOSP build...also with lots of options.
Lastly, if you haven't tried a ROM with an aroma installer, its worth checking out. This rom is TouchWiz and is also very customizable. This one is called
HyperDrive.
Each rom I have mentioned is stable and sweet. I hope I convinced you to try out Eclipse next. I think there is a July 2013 build you should be able to find. Let me know if you need a link. I would need to know which carrier you have. Good luck!
I just recently switched to a Galaxy s4, but I have had a rooted s3 for the past 2 years before that.