Yeah, the upgrades can be a problem on 16GB iPads. My girlfriend had to clear up a bunch of space for the iOS 9 upgrade on hers, though she got the space back after she installed it and the iPad deleted the original install file.
However, I believe there's a way to do it through iTunes such that you don't have to download the install file to the iPad, and can install it on there from the PC instead?
But, when you say 'iPad Mini', do you mean the original iPad Mini, or are you just using it as a generic term? Because you can't install iOS 10 on the original iPad Mini.
Either way, we're waiting until 10.1 before we risk updating.