Glad the thread is coming in handy. It's great so far, I was always running out of space before, but now I have loads of free space, even after reinstalling most of my apps, and the ability to move apps to an external SD card is fantastic (for the apps where the developer has enabled it). I'm also really enjoying the new operating system, although Samsung's contact manager was better than Google's People app, but I've now installed Contacts+, which looks pretty good.
The S3 mini isn't exactly a fantastic phone, so it's still a little slow at times, but the performance is definitely better than it was before I did this.
Just bear in mind that you will void the warranty when you root the phone, and make sure to make backups before you flash it. There is some risk of bricking the phone as well, but from what I understand, it's pretty safe these days, especially if you have backups, even if you don't, you can still download the Samsung ROM and flash that with Odin.
This is what worked for me with the S3 mini specifically. I am using an external SD card to save the install files and backups, should work on the phone memory but I don't know if you'll have space. I have an idea that wiping the system partition won't affect data, but I feel safer having it on an SD card. I also copied my backups to a PC just to be safe.
1. Enable developer options, and select USB Debugging. If you don't see Developer options in settings, go into Settings > About Phone, and tap the build number about 7 times, and it should give you developer options in the Settings menu.
2. Use Kingo Root to root the phone. It's a Windows app, but a nice one click root
3. Use TWP Manager to install TWRP as custom recovery. This will also make you install Busybox
4. Download CM11 and Google Apps zips from http://novafusion.pl/downloads/ and save them to the SD card. Get CM11 because CM12 is still in beta and so is likely not stable yet.
5. Use Titanium Backup to back up all your apps and app data. You can use it again to restore your apps and data after you've flashed the new ROM.
6. Press and hold Volume Up, Home and Power to boot into recovery mode. It should boot into TWRP (Team Win Recovery Project) instead of the stock recovery.
7. Make a full backup of your current system in TWRP. Change the backup save location to the external SD card if you have one. If you really want to be safe, reboot the phone normally and copy the backup to a PC, then boot back into recovery.
8. In TWRP, select wipe > advanced wipe, select System, Cache, and Dalvic cache. (I got stuck in a boot loop when I tried to install over the stock ROM, but it worked when I wiped the system partition first this way)
9. Once the wipe is complete, select Install, add the Cyanogenmod zip, then press Add more zips, and then select the GApps zip. Then swipe the slider to start the flash process
10. A Novafusion installer will run through installing CM11, I kept all the options on their defaults.
11. NB! At the end of the installer, make sure to uncheck the reboot your phone now option. You want to stay in TWRP so it can flash GApps.
12. Once GApps is finished flashing, there's an option to clear cache / dalvic, press that.
13. Once that has done, select reboot system to restart the phone
14. You should see a loader that looks like the image below. First boot takes a while so leave it for a few minutes and it should move onto a status message where it updates a bunch of apps, then it'll start up and go through the first run set up stuff. If it stays on the loader for a really long time, it might be stuck in a boot loop, you can then reboot back into recovery, and try a factory reset, or if all else fails, restore your backup from step 7.
Anyway, after much trial and error, that's what worked for me. Hope it helps