Playing a character with low (or high) scores in Strength, Dexterity, and/or Constitution poses no great difficulties. (Low scores may make things difficult for the character within the game, but as far as role playing by the player and adjudicating effects by the DM, it's pretty straightforward.) Those physical abilities, and all the actions and effects relating to them, exist entirely within the game. There's no overlap between the player's body and the character's.
It's when we get into the abilities of the mind - Intelligence, Wisdom, and Charisma - that the potential for dissonance appears. The boundary between the mind of the player and the imaginary mind of the character is ambiguous at best. Where do you draw the line, and how do you make the mental ability scores of the character matter in play without sacrificing the agency of the player?
In the old Moldvay edition Basic set, there's a pretty famous example of character creation - a fighter by the name of Morgan Ironwolf. Now, Morgan's ability scores aren't half bad overall, but she does suffer from two low ones: a 7 in Intelligence, and 8 Charisma. Not the brightest flame in the candelabra, and a bit abrasive.
Later on, Moldvay uses Morgan in a couple of narratives illustrating how the game is played. The interesting thing here is that Morgan is clearly not played "stupid," nor does the DM ever tell her player, "Morgan isn't smart enough to come up with that." The logical inference from this, and it's one with which I agree wholeheartedly, is not to limit the player's choices at all. If the player can think of it, the character can too. This isn't such a far-fetched way to handle things. People who are overall not too bright do occasionally hit upon some pretty good ideas, and geniuses make some really silly errors. Remember also that the character sees, hears, and feels much more in the setting than you're able to describe to the player and that the player is able to absorb and envision from your description. The player may be smarter than the character, but he's acting on much less information than the character would have arrayed before him. On the opposite end of the spectrum, the fact that players have more time to figure things out than their characters do, and also aren't under the duress of a life-or-death situation, can be considered to balance things out between a player of average intelligence and his genius character.
How, then, do the scores affect the game at all? If they don't, they become tempting "dump stats" if you allow any score-swapping or point exchanging during character creation, and even if you don't, they're useful only to classes that have them as prime requisites for the XP bonuses they confer. Having players roll ability checks to see if their characters know or can do something is one possibility, but not one that I'd want to use. As a player of a low-intelligence character, I'd find it immensely frustrating to be told I can't attempt what I want to attempt because the dice say my character could never think of it. Conversely, it seems unfair to routinely reward a player with information unasked for simply because he rolls under 18 on d20. Fortunately, there are ways to simulate the character's mental abilities in-game without hamstringing player agency.
Intelligence: By the rules, Intelligence affects a character's literacy and the number of languages known. Enforcing these things can go a long way toward making Intelligence relevant to the game. If a player has his low-intelligence PC look at a book, just say, "The squiggles on the page mean nothing to you, and there aren't any pictures," or, "You make out just enough to see that it seems to be a history of the realm, but there are too many big words for you to make much sense of it." Remember, too, that illiterate PCs can't read scrolls, including ones like scrolls of protection that can be used by any class.
When it comes to languages, limit the number of commonly used tongues to 4-10, including a couple human and demi-human languages and a few for monsters, and make sure the party encounters people and creatures who speak them fairly often. Be sure to tell your players which languages are most common, and which are a bit more esoteric, and let them make informed decisions. An intelligent character who chooses one bonus language from a list of 30, without knowing which of them he's likely to use, is getting a very rare advantage, at best. One who chooses from a list of 6, knowing that they may often be useful, has a real edge over his average compatriot who only knows his native tongue.
Finally, role play it, even if the player doesn't. NPCs should react according to their perceptions of the PC. A very smart NPC might naturally tend to address his words to the PC whom he perceives as being close to his own intellectual caliber. A moderately clever but egotistical or insecure NPC might secretly gravitate toward the average and slow members of the party to highlight his own brilliance and shun those who would eclipse him. A halfwit might be in awe of a genius, or might just find him irritating. The possibilities are endless. Play them up, and make your NPCs a little more lifelike and memorable in the process. You might even draw a reluctant role-player into taking a more active approach to portraying his character.
Wisdom: By the rules, Wisdom affects saving throws vs. magical effects. I'm considering expanding it to any other saving throw where common sense or intuition could play a role. Pretty much anything that involves a split-second decision. I see Wisdom, at least partly, as the character's subconscious, processing all the memories of his life's experiences and the subtle stimuli that are beneath his conscious notice. It's that subconscious voice that's going to tell him whether it's better to flatten himself to the floor or duck behind his shield as that fireball hurtles at him. It's his subconscious that's going to register the significance of the faint scraping noise a heartbeat before that axe trap swings out from the wall. It's that uneasy feeling in the pit of his stomach that might have him on guard just enough to look away in time when he steps around the corner right into the gaze of a medusa. Wisdom also represents will power, which is the order of the day when resisting any sort of compulsion, magical or otherwise, that allows a saving throw.
Characters are constantly getting adjustments for actions involving their physical abilities. Saving throws come into play far less often than attack and damage rolls and armor class. I don't think it's overkill to make Wisdom adjustments apply to all or most saves, and makes the ability a little more relevant in the game.
Charisma: Probably the most misunderstood and disregarded stat in all of role playing. I admit that I seldom, if ever, remembered to apply the rules to which Charisma made a difference, back in the old days. This is one ability that can be quite powerful when played by the book, though. Consider that while Strength and Dexterity may affect the chances of each individual attack to hit and do damage, and Constitution exerts an influence over whether or not you'll survive a combat, Charisma can affect whether the combat occurs at all. In Moldvay Basic, when a monster or NPC is encountered, a reaction is rolled on 2d6. A roll of 3-5 is hostile; a 5 will be rolled 4 times in 36, about an 11% chance. That means that, 11% of the time, a +1 bonus will make the difference between a hostile encounter and a neutral one. A roll of 12 on the dice means the monster is friendly. A mere +1 Charisma bonus triples the odds of that happening! Over the long haul, a party with a charismatic negotiator or two is likely to reach the climactic encounter of the adventure in better shape and with more resources than one without, simply by massaging those reaction rolls up a notch to avoid battle or even gain allies.
Of course, Charisma also affects the number of retainers a PC may have at a time, and their morale. Retainer morale is checked at the conclusion of each adventure, and failing the roll means the NPC will not adventure with that party again. A character with average Charisma gives the retainers a base morale score of 7. That means there's roughly a 42% chance that at the end of the adventure, the retainer says, "See ya!" and whatever you've invested in his training and equipment heads out the door with him. Having a 13-15 Charisma knocks that chance down to about 28%; at 16-17, 17%, and with an 18, it's a mere 8%.
A final note on Charisma: One of the strawman arguments I see put forth in favor of "roll" playing rather than role playing is that it's unfair to penalize an uncharismatic player by making him or her role play an encounter rather than simply roll a die against the character's Charisma or some social skill in a skill system. Nonsense! Never, or at least very seldom, should a player be allowed simply to say, "I talk to them," and roll a Charisma check to resolve the encounter. The player should have to decide what to say, and say it in character. It's up to the DM to interpret the player's words in light of the character's Charisma. If the player is socially awkward, stammers and stutters, the DM should take the content of what he says, and imagine how it would sound if articulated by his 16 Charisma character. Conversely, the smooth-talking player should expect to have his character's speech infused with shifty eyes, nervous tics, or a surly tone if he's playing a character with a score of 5.
This rule of thumb can be applied to the other mental abilities as well. A player who explains a concept to an NPC haltingly and in small words, while actually playing a character of 16 Intelligence, might be interpreted by the DM to have spoken eloquently, with proper use of precise language and terminology. A brilliant player speaking for a crude barbarian with a score of 6 might likewise find the NPC reacting as if he had infused his statement with grunts and hand gestures when words fail him.
Of course, if you have dedicated role players at your game table, you'll find them incorporating those ability scores into how they portray their characters all on their own. The dull-witted fighter might speak in monosyllables or always favor the simplest course of action; the unwise thief might be headstrong or reckless; the socially inept mage might be taciturn or arrogant or gruff. If your players aren't so enthusiastic about playing their roles strictly by their ability scores, don't worry about it, and definitely don't give in to the urge to penalize them for "bad" role playing because they're acting smarter or wiser or smoother than their scores might indicate, or worse, tell them, "Your character isn't smart enough to think of that."