Most of the power is used by the lighting - the motors use <10w. I haven't tried solar, but 12V lithium battery packs work well.
We haven't added lithium batteries to our "parts store" up yet on our site. In the meantime: I've tested two lithium packs from Amazon - both work fine, just depends on how long you want them to last between charges. If you search on Amazon for "12V lithium" you will find a wide array of choices. Here is what I tested:
I also have used a bigger and more expensive one of the same brand. It lasts longer, but is more expensive. I was testing on a side table - your Mini take less energy (fewer LEDs) - so the smaller one may be fine. There are several choices that have both 12V (which is what you need) and 5V coming out of a USB port which is nice for charging things. There's nothing complicated about using them - their 12V connection will fit with the barrel connector on the cord that ships with your Mini. Let me know if you have any questions.