I think the important consideration is whether the game allows the player to choose from more than one power which is supplementary to un-powered physical combat, or whether the game is built around a single power that the player does not choose and must get right at the beginning because it is the only method of combat and/or puzzle solving.

In the first case, the best option is to require the player to earn the power(s) within the game. Give the player hints within the game about what powers exist and which direction to go to unlock them, then let the player choose which one to investigate and solve the mystery to figure out what is necessary to earn the power (or the first level of it anyway). If the game is intended to be played through only once, like an MMO, they can gain the power right away. This restricts the possibilities of personal backstory, but you can substitute lore backstory for the faction, mentor, or boss enemy that the power comes from, as well as a bit of current story about the player struggling to learn how to use their new power. If the game is intended to be replayed multiple times, like a single player survival game, the power + backstory can be available for the player to choose at the character creation phase of the next life.

In the second case, it's generally ideal for the character to not know at first that they have a superpower (if they already have it). The player isn't going to know in advance how the superpower works, so since the player has to learn, it feels the most natural if the character has to learn also. In this game structure the power will be strongly tied to the game's theme, and the means of acquiring the power should be also. For example, if the game is about suspicious companies/organizations, then the power is probably intentionally caused by members of that company/organization, thought it might also be inherited but individuals who have inherited the power are monopolized by that company/organization. If the game is about nature or magic rising up against technology or civilization, then the power is probably gained by people who are in a place where nature is powerful, like the woods or the ocean. The power might also be gained by carrying out a specific act to protect or nurture a piece of nature, thus earning the positive opinion of Gaia or whatever the spirit of nature or magic is called in your setting. If the theme of the game is about an ordinary person trying to compete against special people, then the character would probably be crafting the tools of their power, like Batman or Ironman. If the theme of the game is about history and lost traditions, the power would be taught by a mentor, probably one of the last survivors of the past world where the tradition was still powerful.