Doubled dies are created during the die production process, before the coins are minted. To make a die, a blank die is heated up to soften the die metal. Then under heavy pressure it is forced against a hub containing the coin design. After being allowed to cool to harden the die, the process is repeated multiple times. The die is heated again, pressed against the hub, and allowed to cool. After several cycles the design is completely imprinted on the die and it is ready to mint coins.
If during the die production process, the die is not perfectly aligned for each squeeze against the hub, then a doubled die is created. Part of the image on the die is doubled. If the die is not heated evenly or is not allowed to cool completely, then the die may become warped and the next impression against the hub will not be perfectly matched. Again a doubled die will be created. These and many other things can go wrong during the die creation process resulting in dies doubled in one way or another. For a taxonomy of types of doubled dies and more detailed information on the minting process, see The Authoritative Reference on Buffalo Nickels. Note particularly the Class IX variety.
Re-punched mint marks (and over mint marks) are created in a similar fashion to doubled dies, although on a much smaller scale. For dies being sent to the Denver and San Francisco mints, the reverse dies needed to have the mint mark punched into them. The die would be secured, a punch positioned, and the punch struck sharply with a hammer. This process was repeated to create a solid impression of the mint mark. If the punch was not aligned precisely the same each time before it was struck or a different punch was used that perhaps had a different sized mint mark, then a re-punched mint mark would be produced on the die. If a punch with a different mint mark on it was used for a strike, then an over mint mark would be produced on the die.
Punching the mint mark focuses a lot of energy in a small area of the die. This makes it somewhat more likely the die will be weakened or damaged in some way, either visibly or internally. This is one factor that contributes to die life (or perhaps I should say die death). Because re-punched mint marks likely involve hitting the die more times (and arguably harder), the dies for RPM’s may have shorter lives than other dies. It also may be why you frequently can see minor die cracks near mint marks.