iPad mini ramdomly shutting down or getting a full blue screen

This is usually a sign of the flash memory chip which is basically the hard drive of the iPad that is going bad. If there is no information that needs to be backed up on the iPad I would recommend doing a DFU restore of the iPad and see how that goes.

Here is a guide for restoring apple devices:

https://help.ifixit.com/article/108-dfu-...

If it does end up being a hardware fault this would require board-repair / micro-soldering work done to the iPad which would in my opinion not worth it as the iPad mini 1 no longer receives software updates (iOS 9.3.5 last version).