What type of box?
Howmany times/how easily do you want to remove the Arduino Mini?
Hot glue is the way to go in many occasions.
- It does not conduct electricity, so is pretty safe. (Please check your specific hot glue first)
- It sticks quite well, for quite a time. You should also check if it's water resistant or fit for your target environment though.
But, it may have some side-effects:
- It can get quite messy (which does not look really professional)
- It's not that easily removable and has to be "re-glued" when you want to stick it back.
- It might be too sticky (in a carboard box, you won't be able to just rip out the Arduino, withouth damaging the box).
You could use double sided tape, which might actually work fine. Or atleast better/neater as the hot glue.
A neat solution would be sticky velcro. You can easily attach, detach and re-attach the arduino to the velcro. It looks quite fancy and should hold well.
Again, you should check the specific velcro, it might not last 1000+ times of re-attaching (become loose or smh). But will be a good option overall.
You can use mint-boxes as an enclosure. But they're usually made of tin or another conductive material. Please take care and cover the back of your arduino (or box) with electrical tape, to avoid shorts.
Extra-extra, you should (for "replacability") add connectors or headers to your Arduino. Since soldering the leads directly will still give the need to un-solder or resolder your Arduino when replacing.
Note on making your Arduino project a consumer item
You might save yourself quite some money by using a barebones arduino. Or actually, an ATmega 328P chip. Which is around 3€.
These will be a little more difficult to program, but might also look more professional.
You can also enable "program protection" fuse bits, so that others can't download the code from your arduino/microcontroller.