Given that the DMG has rules for custom backgrounds, a willing DM can work with a player to "customize" with the bare minimum of effort by simply swapping a attribute, skill or feat from one of the prefabs, without having to create a whole new background. In my game, I'd just ask the player to justify it.
Player: "I'd really like to take the Acolyte background for my cleric, with a +2 to Wisdom, but can I put the +1 into Con?"
DM: "Why would your character have received that benefit?"
Player: "They studied meditation during their training, able to keep their body fit even when going without food or water for extended periods."
DM: "Sold!"