Can my mac mimi support software above 10.6.8?

Yes, with sufficient memory you can go to the current El Capitan

http://www.apple.com/osx/how-to-upgrade/?cid=wwa-us-kwg-mac

You Mac also support the next verion, Mac OS Sierra.

Although Apple say 2 GB is the min memory required, you will not be satisfied with only 2 GB. Even 4 GB of memory results in marginal performance. Your Mac support 16 GB (2x8). This place sells compatible memory

https://eshop.macsales.com/shop/memory/Apple_Mac_mini/DDR3

Here are requirements for El Capitan: OS X El Capitan - Technical Specifications

For Mavericks it isn't much different: OS X Mavericks - Technical Specifications

If your Mac Mini has 2GB of memory it should be enough, but I would recommend upgrading to 4GB for better performances.

I would recommend 8 GBs as a minimum.