To me, it sounds like the LCD is out. You could buy the LCD replacement but it would be more work than needed. If it were me in this situation, I would purchase a used Galaxy s3 mini of the same model as yours (since they are selling for around $30 to $40 online) and swap out the motherboards. That would allow to keep all your information and would only take about 10 to 15 minutes to swap them over.
If this is the route you take, and you are still having the same issue, then the problem lies on the board.