iPad mini home button replacement cost?

I would assume that the Home button replacement would fall under Apple's fixed repair rate of $199 (out of warranty)

https://www.apple.com/support/ipad/repair/other/

In the interim of replacing your Home button, you can use AssistiveTouch which floats a virtual button on-screen which can mimic all hardware buttons on the device, including the Home button.

I personally use AssistiveTouch all the time, just to prevent wear and tear on my Home button...I seldom use the actual hardware Home button...you can even do a device reset (press & hold both Home and Sleep/Wake buttons for ~15 seconds) using the AssistiveTouch button if your Home button is bad (but not if your Sleep/Wake button is bad).

Settings>General>Accessibility>AssistiveTouch: turn ON

*Note* If you power off the device using the AssistiveTouch panel (tap the on-screen AssistiveTouch button, Device>Lock Screen (tap and hold to power off)), you will NOT be able to power on the device if your Sleep/Wake (power) button is not functioning. You will need to connect the powered off device either to a PC's USB port or an external charger to power the device back on.

I was told that you can apply rubbing alcohol, gently rub around the button. This may clear dirt and the button will start functioning again. I didn't have any rubbing alcohol, but I used a piece of tissue soaked in vodka. It failed the first time, but later I decided to try again and it actually worked. I now have a fully functional Home button once more.

Hope this helps people with the same problem.