My first advice is to increase the RAM on your Mini, 2 GB is the minimum amount needed to use El Capitan and you should install at least 4 GB RAM for it to work at reasonable speed. RAM is quite cheap these days.
As for the problem there are some things you can try:
1. Restart Mac while holding Shift key until you see the login screen, then login normally and try update again. It will remove caches, temporary files etc. which might cause the problem.
2. Next step is Terminal because Apple unfortunately doesn't provide standalone downloads of Safari etc. software, if you have no experience with command line it could be somewhat scary. You can find Terminal with Spotlight (upper right corner, magnifying glass) type terminal into Spotlight.
Copy the lines after : mark to Terminal. First you check what updates are available: softwareupdate --list
Then you install updates: sudo softwareupdate --install nameoftheupdate
Alternatively sudo softwareupdate --install -a to install all available updates
If you install update which requires reboot such as 10.12.6 update (Safari may or may not require reboot) first wait until Terminal notifies you to restart Mac, then use sudo reboot command and wait until Mac completes install & reboot.