I've stood by and watched a lot of characters get rolled. I've watched the reaction to the dice and the process of settling where the ability scores get placed, what equipment gets chosen and the eternal questions of race, class, skills and so on get answered. Even after all this time, I rather enjoy this part of the game, even though I haven't rolled up a character to play myself in, oh, about eight years.
My core method of starting a character hasn't changed much since the early 1980s. I have the player roll four 6-sided dice, discarding the lowest die. These are then placed according to the player's will under the six familiar stats (do I have to list them?). The player does this while keeping the minimum requirements for classes in mind. This used to be understood by everyone that played, but I suspect that's changed, and that 5e no longer requires minimum stats to be met in order to be a ranger, a paladin, a monk and so on.
I'll get into the question of why minimum stats are important (eventually), but let's start with why using a 4d6 and not 3d6 matters. From the beginning, I bought into the argument that player characters ought to be "better" than ordinary people. Not because they're heroes or because they uphold some great cause, but just because a better collection of stats ensures a recognizable edge on NPCs without those stats. We can take a simple depiction to express the value in this: a player is running across a rooftop, being chased by an NPC. The player has to make a leap from this rooftop to that; the distance is part of the calculation, but so is the player's strength or the player's dexterity (depending on whether we feel the ability to cross the distance is important or the ability to land well on the other side). The player makes a roll and then the NPC makes a roll.
We would, I think, rather play in a world where the likelihood is that the player will make the jump and the NPC will fail. This fits with our dramatic instincts and with our self-image. Whether or not we do good things, we are the good guys (we always think everyone else is the bad guys). It just seems right that our chance of making that jump ought to be better than someone else's. And when we make it, and they don't, the world seems to be operating in good order.
Of course, we could miss. And of course, the NPC might also make it. But the number of times this departure from our mental projection occurs matters to us. Too many departures and we'll start to feel the game is rigged somehow, that there aren't enough chances for us to win. Again, we're the good guys. The DM's NPC doesn't really matter, right?
There are many who feel it's right if the players have exactly as much power to make that jump as NPCs ... but I can't agree. As a DM, I don't have near as much invested in a NPC as the player has in their character. It takes very little for me to produce another NPC as needed, as I don't need to spend a lot of time determining all their stats, their equipment or their particular details ~ certainly not for an NPC chasing the character across some rooftops. The difference is considerable. And since the game is about the player, I'm fine with the game interface being balanced for the player.
The 4d6 seems to make a good average improvement, without that improvement being excessive. I've seen campaigns where it was 6d6 and drop the bottom three dice, or eighteen 3d6 rolls, or some other combination, most of which were named in the old, old DMG from 1979. Other systems seem too balanced towards a superior player or they seem unnecessarily time-wasting. 4d6 less one die gives an average of 12-something instead of 10.5. That is just enough to matter, without being enough to make the player feel safe. It fits with the amount of play I want the player to have within the game's structure.
The next question, then, is why let the character select where the rolls go? There is a top-to-bottom philosophy that insists the die roll order fits the order of stats. I think I understand: DMs felt the players shouldn't be allowed to consistently run the same class of characters all the time. "Why does Glenn always have to play a fighter?" goes the logic. Yet I think this is a DM's problem. If it were Glenn's problem, Glenn would stop playing a fighter. I think we should stay away from one person imposing their problem on another person's contentment. I don't think there's a strong argument to be made for such things.
I like that the player gets some control over what they have to run for what I expect to be a long time. I don't start games that are meant to stop soon and I haven't found many players whining about having to play a cleric or a mage because that's what they chose two years ago in real time. That may be mitigated by my henchman rule, but I don't remember it being a problem before. If it is a good game, and if all the player classes can prove themselves to be relevant in tackling the game's interface, then I think players just like the fact that they're stronger and tougher and have more resources at their disposal, whatever their class is. The problem arises, I think, when the character improves in level but no real change results as to what the character can do. The fighter was always an issue here: more hit points and a better combat table, even more attacks, does not make for a dynamic, growing character. Thankfully, I've solved that problem too.
The player, then, reorders the numbers around the character class requirements and around what obstacles the player intends for the character to overcome. When considering the number under a given stat, the player thinks in advance, I'm going to be solving a lot of problems that need an effective intelligence or constitution or charisma. I should then start building my character's goals around my strongest stats.
Hm? No? Don't look at it that way? Well, you're not alone. No one looks at it that way. They think, a high stat under charisma will make people like me, a high stat under strength will mean I do a lot of damage, etcetera. They don't build characters ~ or agendas ~ around their stats. But they ought to.
Consider. I start with six rolls (and I'll roll these out). I have a 14, a 15, a 10, an 18, a 13 and a 9. Swear to gawd, I just rolled an 18.
Could be a fighter, but there's only a little Con and not much of anything else; I'd rather try a cleric. It works better for this example, anyway. I won't get a wisdom bonus in spells for ages but I'll put the 18 under wisdom and the 15 under constitution. I feel I'm going to want to run a congregation someday so I'll shift the 14 under charisma and I'll take the 13 under strength so I'm able to carry armor and weapons without getting slowed down much. That's a 10 for intelligence, but I plan to be a bullheaded cleric anyway, and a 9 for dexterity. I doubt I'm going to run over many rooftops. I'm far too wise to get into that sort of situation.
I'm running with a DM that is very much like me, so as often as I can I'll try to build up a store of knowledge that will allow me to make a wisdom check when the situation seems unclear. My DM feels that someone with a high wisdom ought to have a good sense about the motivations of NPCs or the best way to go about getting help from people and organizations. Past that, my high constitution will mean I'm not afraid to enter into hard, difficult to survive places. I won't be foolish about preparing for icy or sweltering climes (and my wisdom will give me checks there) and my constitution will give me a good, tough ability to hunker down when the weather and the terrain turns sour.
I'm not planning on influencing a lot of people with my charisma, at least not until I'm in a position of power, because I don't feel I want to lie much with this saintly man of the cloth; but I do want to be left alone. A high charisma will tend to be treated among the better citizens and as I intend to keep myself clean and well-groomed, the charisma ought to help me if the rest of the party (or strangers in the area) start acting like a bunch of louts. My strength is at least still above average ~ but not too much. I'm going to be getting into the fight until I amass enough spells to be an effective caster, so I'll need good armor. I guess I can take on a fair burden, too, since I'm going to be human. That makes me a big, burly member of the party among these elves and half-elves, but I don't mind carrying a good, solid load. If I move at a rate of 3 hexes per round, I should do all right. Not a chaser, but good enough to stand in front of a mage.
Now, that intelligence. My DM usually does intelligence checks when players take an action that is flat out stupid; I think with my experience at the game that I'll take my chances. 'Course, I'm won't be the brightest penny in the box where it comes to tactical games or giving orders; guess I'll let some other lead the army we accumulate someday, else I'll walk them straight into a rout. And with my dexterity, I should be fairly deliberate about keeping my feet on firm ground, walking on the side away from the cliff and vying for knowledge that doesn't require much hand-to-eye coordination.
That, finally, gets us around to why minimum stats for classes are a good idea. But, this post is already pretty long. I guess I can manage that for another day.