Ok, I know it's a big discussion between all the stats determining systems. To be honest I've always been rolling for stats and never considered switching to point buy or standard array as I find it boring. 
I've been thinking about that lately as I will be DMing my first longterm (hopefully) campaign and want to do this right. I've been doing some tests using the rolling method of 6x 4d6 thinking that if I can get a system in which my party's stat totals were in the same range, that it would be a fair one. Long story short, it didn't go very well as I landed on a system that needed sometimes tens of rerolls in very bad luck cases (too high or too low totals). 
Now, taking a look at point buy, that's basically a better version of my flawed attempted system. The only thing missing for me is the opportunity to get a very bad score which, I think we can all agree, for some characters is a very fun RP tool (low INT, low CHA for instance - and let's not forget the famous "What did you roll? ... -1", always a banger). However that is something that you don't get by using point buy.

Fellow players and DMs, what would you recommend? Should I just permit my players to take a 7 or even a 5 in a stat if they want to? If so, should they get anything in return? As a (kinda) experienced player, I would love to RP a 5 or 7, even if I don't get any benefit from not taking an 8 in that stat, but is it fair to the players? I don't see how it's not. 
On the other hand, should they get a bit more points added to the 27 by taking a very bad dump stat so they have something to compensate?

I'd allow a -2 to a score to increase any other by +1. 

I don't like rolling for stats because someone might roll much lower stats than someone else and it might affect the fun they have at game time.

I create LOTS of builds in my free time, with no intention of ever playing them.  I start with attributes of 14, 14, 12, 12, 10, and 10.  Then, I modify the attributes with any modifiers applied by the PC's race or feats.

The most I'd consider allowing would be dropping one stat further to get more points on a 1-for-1 basis, and you still can't buy a stat above 15 to start. 

Why?

Because you're creating the dump stat scenario, where players bury the stats they don't think they need to get more powerful, and you get weirdly lopsided characters. Point buy already allows 8/8/8/15/15/15, which I can't say I love.

I have been in games where a player has played a very unintelligent character with int 8 (it is also fine to play a character with normal intelligence but a bit less "book knowledge" thant average with int 8. (Ditto for other stats).

I would have no problem with a player choosing to replace an 8 with a 7 for roleplay reason and get nothing back in return.  If I did offer something back I would make sure they are not gaining overall, they are taking a super low stat for role play reasons not to increase power. 

If Con, Dex or Wis is the dump stat I would be more generous, all these saves ar every common, Con helps with hit points, dex helps with initiative (I am assuming they character does not rely on dex for AC). Having something like Int as a dump stat doesn't reduce your effective very much nearly all the time (there are exceptions such as a mind flayer encounter). For example a monk gets more benefit from single point rise in Dex, Wis or Con than 2 or 3 points in Int (or charisma and maybe even strength).

There is a floor however, a creature I would probably set it at 4, below that level they would not be accepted into any adventuring party.

• A low strength character would struggle to carry all they need for an adventure
• A low dex character would be too clumsey and slow
• A low con character would be unwell an not up to the vigours of advenuring
• A low intelligence character would not be able to understand simple instructions
• A low wisdom chracter would be unaware of te world around then
• A low charisma charactre would be unable to communicate effectively.

Quote from The_Crycket >>

One of my DMs gave everyone the same starting stats.  We could all take a -2 in one to bump +1 in another for up to 2 stats. At some point really low stats can cause an issue in the DM makes you follow them.  Have a great idea as a player? Sorry you character is not smart or wise enough to have it.  Want to have some armor and belongings; your character is encumbered and slows down the entire party.  Really low CON, you get sick often and have to do a check to stay healthy.  

 
I will say I hate DMs like this. It is too much like penalising players for -- stats? Nah. That just doesn't sit right with me. It's D&D, It's not a Sim.

One method I quite like that gives players the flexibility to have low scores if they want, while still producing sets of stats that are generally a bit better than the standard methods, is this:

1. Every player rolls the usual 4d6 times six dropping lowest
2. If there are fewer than six players in the party, the DM rolls additional sets of stats to make up the difference
3. These numbers then get placed, in the order they were rolled, in a 6x6 grid on a spreadsheet
4. The players can then choose any row, column or diagonal, and use those six numbers for their stats. Each grouping can only be used once, however

This generally produces a couple of really good sets, for players who were looking at MAD builds, but also provides the option of taking a set with one low score if they want it for RP purposes

Honestly, I think 8 is low enough for pure roleplay; as has been said, part of the reason current point buy has a floor of 8 is to limit dump-stat optimization. You can reasonably roleplay- for instance- 8 INT as anything from a Grog expy to just competent but not particularly book smart/analytically minded. Taking an excessively low score for roleplay functionally means you’re handicapping yourself for the performative value, except the roleplay itself is supposed to cover the performative side of things. Honestly, if the table’s okay with it you can try just letting them take a lower score without getting any boosts elsewhere, but really I feel like this is more a case of just telling them to play their characters how they want, within reason.

We used to go with 4D6 drop the lowest, roll all 6 and place them in any stat you want.
Then you can do a single slight/favor change -2 in one stat and +1 in another.

The Dm gets the chance to let you reroll everything(start over)  if the character is deemed unplayable. Essentially if your score total was below 70-72 you would get a restart.

If you’re looking to roll and keep everyone on par, I’d say either go with AntonSirus method, or you have each player roll a stat line, then all together choose and use the same one. They’ll put the numbers where they like, but all have the same 6 numbers.

I’m just putting this out there because it seemed like fairness was the bigger problem the OP was trying to solve. 

I’ll also add, imo, fairness of this sort doesn’t really matter. I’ve been in games with people with wildly different stats, and it was just fine. And I say that as someone who’s been on both the higher and lower ends. Will you really notice that the 18 str fighter hits more often than targets save against the 16 int wizard? You might notice two side-by-side melee classes, but even then, the number of rolls needed to really make it matter gets high enough that without breaking out an excel sheet to track it, it doesn’t come up. 

I've never heard of that method.  I like its fairness.

You could try another alternate method for rolling stats. This method is best for groups with 3 or 4 players, but can still work for groups with as many as 6 players. You go around the table and each player rolls 4d6 and drop the lowest. You go around the table like that until you have 3 sets of stats generated. Then let the players choose one of those sets by consensus, and every player uses the same array, but places them however they want among their 6 ability scores. Just in case that wasn’t clear, I’ll give an example* with 4 players since that’s typical for most tables.

Set #1                                             Set #2                                             Set #3
Player A rolls: 10  Player C rolls: 17  Player A rolls: 16
Player B rolls: 17  Player D rolls: 12  Player B rolls: 10
Player C rolls: 13  Player A rolls: 14  Player C rolls: 11
Player D rolls: 12  Player B rolls: 17  Player D rolls: 11
Player A rolls: 10  Player C rolls: 13  Player A rolls: 9
Player B rolls: 12  Player D rolls: 15  Player B rolls: 11

Then all 4 players vote for which of those three sets to use, simple majority wins.

(*My dice luck is notoriously bad, so all of those 3 sets might suck eggs. If so I apologize.)

Quote from Grankausto >>

...
I've been doing some tests using the rolling method of 6x 4d6 thinking that if I can get a system in which my party's stat totals were in the same range, that it would be a fair one.
Long story short, it didn't go very well as I landed on a system that needed sometimes tens of rerolls in very bad luck cases (too high or too low totals). 
...

 
Search online for "D&D 5e Random Stats with Cards".  There's a number of homebrew stat generation methods out there using decks of playing cards to generate random stats that all total the same point value, i.e. 72.  I don't remember a specific system well enough to recommend it, but with some Google-Fu you can find a system that guarantees every player in your game has the exact same BST while still having totally randomized distribution of stats.  Look up one that sounds good to you and enjoy random stats with fixed stat totals.

I don't see the need to do the -2 / +1 for flavor when just about every race gives you +1/+2 or +1/+1/+1

Its to help random low rolls, not for flavor.

You can always make your own array and give them that to play with. You can even stipulate that the highest score can not be in the Main stat, or other parameters that you like.

It's already been said, but I'll add my vote - an 8 is low enough to roleplay that you're bad at something. Roleplay is separate from mechanics, and mechanics can be emphasized or de-emphasized to support that RP concept. No one is going to break down your door and tell you that you can't act dumb as a box of rocks with an INT score of 8.

Stats stay in a certain range so that the game will function as intended. You don't mess with that unless you absolutely have to, and you really don't have to in this case.

So to be clear, you don't need a 5 INT to play dumb. You don't need a 6 DEX to play clumsy. A stat floor of 8 is not depriving anyone anywhere of any roleplay opportunities, only the mindset that mechanics are the end-all-be-all of roleplay are depriving people of roleplay opportunities.

Thank you all for your answers, I think I've made up my mind on the system I will be using after reading all of you.

Point buy is the way to go for me, since, as you all pointed out, it's more balanced for the game (I'll be doing my own homebrew adventure but it's the simplest way to have balance between my different PCs). 
Once the 27 points have been attributed, I will be allowing a -2 dip in 1 stat (up to 6 if it's a stat at 8) in profit of a +1 in another stat. However the +1 cannot put the stat over 15 before racial bonuses. That will keep the balance by not allowing a higher ceil in the stats but allowing for really low stats if any of my players want it for RP reasons. In any case, that will be discussed with my players to make that clear.

Finally, I've been considering allowing a free feat for all. I've been doing some research on that as well and since it would be a feat for each PC, that shouldn't break the in-party balance. However that would allow the stats to be as high as 18 after racial bonuses if taking a half-feat, which I'm not sure how I feel about. 
I've seen some people saying "Just ban the feats you don't want". But the half feats are the ones that could be considered "flavour feats" like actor, chef, observant, keen mind and all racial feats and banning them would mean that I'm forcing my players into GWM, SS and the like, which I don't want to (not saying I'm banning these as 1st level feat, just that I don't want them to be the only one remaining). I think I will allow all feats tho, some characters will indeed get the 18 at level 1 and the 20 at level 4 but I will manage the encounters accordingly.

Anyway, still have to do a bit of thinking on the feats but I think I have the ability scores down. Thanks for your advice!

You can always say that if they take a half-ASI feat, their racial bonus would be limited to just +1/+1 or +2. This way they still have the same ASI in total plus the feat's other benefits, keeping it balanced.