If you don't feel confident in doing the repair then take it to a reputable shop and have them do it for you. The iPad mini is not for the feint of heart when if comes to screen replacement. From backlight issues to phantom touch, there are many things that can go wrong if you aren't careful. If you are interested in device repair, have a knack at it and not too concerned about things going sideways or are willing to soldier on when they do, then have at 'er.
I have an iPad 2 here that will boot up , the apple logo comes on, then disppears. About a minute later the home screen will appear and it functions as normal. Is it possible that you may need to wait a minute or so after booting the customers iPad?
It's likely that the iPad will need to have the lcd cable reconnected. Sometimes smacking the back doesn't work. With the proper instructions And patience you can open the iPad and do this. Be sure to power the iPad off first though. Just hold the power button for about 5-10 seconds then swipe not the screen towards the top where slide to turn off would be. You should see the backlight dim slightly then turn off. Follow the ifixit guide to open an iPad 2, and you'll be good to go.