Rolling ability scores is a time-honored tradition across many editions of D&D. However, it can sometimes cause problems for players and/or the DM. For example, one player character may end up much weaker or much stronger than the rest of the party, which can result in a poor experience for some of the players. In other cases, a player may have their characters repeatedly commit suicide-by-monster so they can try to reroll for higher stats, which can be quite frustrating for the GM and other players.
A lot of character power is dependent on ability scores. How much time a player gets in the spotlight can be heavily impacted by how powerful their character is. Having one character be much more/less powerful than the rest of the party can result in imbalanced time for the player of that character in the spotlight, which results in players having less fun. Giving all PCs fairly equivalent ability scores can help avoid that problem.
What approaches are available to mitigate these problems?
Note: Answers should ideally be able to prevent both the "Joe rolled all 7s, and his character is useless" problem and the "Karen rolled all 18s and her character makes everyone else's character useless" problem. That is, an answer that only avoids very low average/total scores is not as good as one that avoids both very low and very high average/total scores.
I intended this to be a canonical question for all editions. I don't mind edition-specific answers as long as they're not also character-specific.
Best Answer
Option One: Non-Random Ability Score Generation
Rather than giving up control of how powerful player characters are to the whims of fate, you can instead use systems that attempt to consistently produce ability scores at an established level of quality. The two most popular schemes for doing so are standard arrays and point-buy systems.
Standard arrays present players with between one and three pre-generated sets of ability scores. Players select a set and assign the scores within to their abilities however they want. For example, players may be given two arrays: 16 14 14 13 10 8 and 16 14 13 12 11 10. Carol wants to make a fighter, and chooses the second array. She then picks the 16 for her strength, the 14 for her constitution, the 13 for her dexterity, the 12 for her wisdom, the 11 for her intelligence, and the 10 for her charisma. Bob wants to make a wizard, so he chooses the first array, and assigns the 16 to his Int, the 14s to his Dex and Con, the 13 to his Cha, the 10 to his Wis, and the 8 to his Str.
Point-buy systems instead give the player some sort of base ability scores and then a pool of points they can spend to increase ability scores. In some systems each point you spend increases an ability score by 1, while in others getting a single high score costs more than getting two medium scores. The Pathfinder point-buy scheme is an example of the latter. Point-buy systems give the players more control over their characters' ability scores, which generally makes them happier, but it can also increase the potential/temptation of heavily min-maxing characters (e.g. a fighter with 18s in Str/Dex/Con and 6s in Int/Wis/Cha), which can still result in some frustration for the DM.
Option Two: Collective Ability Score Generation
Most problems that arise from rolling ability scores ultimately center around large differences between player characters. We can resolve this issue, and still roll ability scores, by stealing the idea of having a set of arrays from the first option. Instead of each player rolling an array of ability scores for their own character, each player rolls an array of ability scores for the party and then chooses any of the rolled arrays to use for their character. Once generated, these arrays should be saved and used again for any characters created later, rather than rolling an additional array.
For example, Alex, Betty, and Charles are making character's for Dana's new campaign. Alex rolls 6 11 8 13 9 10, Betty rolls 18 7 12 11 15 10, and Charles rolls 16 12 14 13 15 9. Alex and Charles decide to use the array that Betty rolled, so they can put the 18 in their main ability score, while Betty opts to use the array Charles rolled so she can make a more MAD (Multiple Attribute Dependent) character. When Alex's character dies a few levels in, he doesn't roll a new set of ability scores for his new character. Instead, he goes back to the three arrays generated when the campaign started and chooses one of them. If Eric joins them a few levels after that, he also would use one of the same arrays everyone else picked from when they created a character.
This approach means that if a single player rolls poorly, they're not stuck with a weaker character. If a single player rolls well, everyone can make characters who are just as strong. While this approach tends to slightly increase average party strength, having all the characters on an even playing field makes it easier for the DM to adjust encounters appropriately.