I'm not sure about this but you could try peeling off the homebutton cable with the help of a heatgun . But not the whole thing up to the soldering point.
Just lift the area where the homebutton is and see if it the culprit is the homebutton itself, or maybe dirt around it or if its the golden plate with the black dot in the middle.
Normally on the iphone 4(s) a malfunctioning powerbutton (yes not homebutton) has a faulty golden contact plate to be precise the black plasticdot in the middle has broken off or was pushed so often that it got flat. The homebutton on the ipad mini looks like having the same mechanics.
For the iphone's powerbutton you could order just this gold contact with the black dot and adhere it onto the cable. I saw people selling only the contact on ebay for a dollar.
Maybe its the same size and fits for the ipad mini too.
Otherwise you could order a new cable assembly and peel off the contact.
Or if you are not afraid of soldering you could just exchange the whole cable with the ordered part.