You'll need either a full Linux installation or a live USB with at least 4GB of free space. Google "pendrivelinux" and the first result should be an entire site dedicated to telling you how to make a live USB. I recommend Ubuntu, it's easy for new people to use.
- Open a terminal (if you're on Ubuntu, type terminal in the search box)
- Plug in the internal SD card from the Kobo Mini
- Type "sudo fdisk -l" (That's a lower case L for people with silly fonts)
- You should see a ton of stuff, most importantly, each section should have a header like "Disk /dev/sda: 64.0 GB, 64023257088 bytes, 125045424 sectors". Look for one with "Disk /dev/sd[probably the highest letter]: [Size of your SD card]..." and note the letter, it's important.
- If you're on a full blown installation, you're about to create a 4GB image of the card, so cd to somewhere useful. If you're on a LiveUSB, look through the results of "fdisk -l" for the /dev/sd[letter][number] that corresponds to a data partition on your hard drive (should be quite large and have a ton of blocks) and type "sudo mount /dev/sd[letter][number] (for example, /dev/sda1) /mnt" then type "cd /mnt".
- Type "sudo dd if=/dev/sd[letter corresponding to your SD card] of=kobosd.img bs=4M" then wait a bit. It won't show any progress but it's copying 4GB of data from your card.
- Eject the card from the kobo and insert the new card (of whatever size) and type "fdisk -l" again. Look for the letter again. It will possibly be the same letter that the Kobo SD had.
- Type "sudo dd if=kobosd.img of=/dev/sd[new letter] bs=4M" and wait again. It'll probably be slower than last time since you're going to the card instead of from.
- Type "sudo fdisk /dev/sd[new letter of SD"
- Press "p" and the partitions should be printed. Make a note of the start value of the last partition and copy it or write it down somewhere. Also the number after "/dev/sd[letter", that's important too.
- Type "d" and when prompted, enter the number you just chose.
- Type "n" and when prompted for type, just hit enter. Then enter that number again. When it asks for a starting position, enter the start value from before. When it prompts for an end value, just hit enter, this should use the maximum available space.
- Press "t", "3" then "b" and hit enter.
- Type "w" then "q"
You're done! You can put the SD back in the Kobo Mini.
I've only tested this with my Mini and a 16GB Class 10 Patriot MicroSDHC card.
If you screw up during the whole fdisk thing, just hit q and start again. Nothing is modified on the disk until you press w.
To get the SD out of (and into) the Mini, all you need to do is pull off the back cover and unscrew the plastic cover behind that. The SD slot is rather obvious and easy to use.
EDIT: The resize method I listed above may have issues with the Kobo picking up the size of the new FS. Ignore the 'sudo fdisk' line and everything below it. Instead run 'sudo gparted' from the console (should be preinstalled on Ubuntu), then select '/dev/sdX' in the top right where 'sdX' is the same one you've been using the whole time. Then right click the bottom item (the label should be 'KOBOeReader' in the list that shows there and click 'resize/move', set 'free space (both before and after) to 0 then click 'resize/move'. Then just click Edit->Apply all operations. It'll give you a warning but that's fine. It's a much easier and technically superior way of doing things.
If you have done the instructions I already put there and find your Kobo isn't detecting the free space, you can put the SD back in, type 'sudo gparted', click the top right thing and select your card (the size should make it recognizable) then right click 'check' then Edit->Apply All Operations. The Kobo should pick it up after that.