I perused the 2nd edition Dungeon Master’s Guide last night, and stumbled upon a passage discussing the difference between using 3d6 to generate a character's ability scores and "4d6 drop the lowest":
Some of the core 5th edition mechanics were developed in Mike Mearl’s basement in 1981 as a complement to the Basic Rules. Hence, the above quote applies to 5e, even though it wouldn't to other editions. Because 4d6 drop the lowest is principally used in D&D 5e, the authors of D&D 5e intended characters to be epic, not average.
If you want to draw your characters from a pool of average people, use 3d6 for your ability scores. Mechanically, the average roll is 10.5 for 3d6 (as opposed to 12.2 for 4d6 drop the lowest), so the odds are stacked to make a weaker character.
Why do this at all? Isn't D&D about Epic People with Epic Chins doing Epic Things?
If you want a game about epic people, then use 4d6! Please do.
However, there is something romantic about normal everyday people who take up the spear and face the utterly terrifying darkness of the fantastic world. Heroes are supposed to do this, but for everyday people it is an act of extreme courage. Using 3d6 makes for a more interesting game and is perfect for a dark and gritty campaign. It evokes an OSR flavor.
You should obviously discuss the tone of the game with your players. Most of my players eschewed the idea. They believe heroes are people who are a cut above the rest (one player argued that her character was an epic person failing to live up to his potential). So we won't be using this option for our current game, but this is something I'll keep in my tool bag for later.
5th Edition Example: Bad Scores don't make you completely useless.
I rolled a character using 3d6 for a recent game. Enter Josh, first level human fighter. He is a street rat with the courage to stand up (and die) for justice. Josh wanted to be a magician, but never had the brains or wisdom to cast spells. After he was kicked out of every wizarding college, he was forced to survive on the streets. Here are his stats:
Str: 4 (-3)
Dex: 13 (+1)
Con: 12 (+1)
Int: 8 (-1)
Wis: 6 (-2)
Cha: 10 (+0)
This is after racial adjustments. He couldn't carry his starting equipment without medium encumbrance. My DM floated me warnings about having such a "weak" character.
But josh is not as meek or hopeless as you might think. He took the magic initiate feat so he knew one healing word spell, which despite the negative wisdom modifier, had a minimum of +1 hp. That instantly wakes up a downed character as a bonus action, not too shabby. He also knew the spare the dying cantrip (you learn a few things even if you fail out of magic school).
Josh also took the archery fighting style as a first level fighter, which got him a +5 to his longbow attacks. His background gave him a proficiency in stealth too. His is still a useful person despite his -4 total ability modifier.
Had he lived, josh would have probably multi-classed in to rouge, but who can say. He died after rolling two natural 1s on death saving throws. Nobody comes back from that; epic hero or average adventurer.