Cards with more storage tend to use SDHD SDHC which the mini can't read.

The 2g cards don't usually need that and just use SD.

Strangely I use a card with SDXC and that has been working for me.

Edit corrected spelling and added link for the card I use: https://www.amazon.com/G-Skill-Micro-Memory-Adapter-FF-TSDXC64GA-U1/dp/B00CGY9VO4