I slept on it, which meant that I rolled the idea around in my head a little, and decided to use this post as a sort of sounding-board. If you have any ideas or complaints or anything at all to say, let me know in the comments!
Attributes serve a couple of purposes in 5e:
- Character specialization. A 15 strength fighter is different than a 15 strength druid (but more on this later).
- Multiclass restriction. Since your character is not good at everything, there optimal and suboptimal classes. (Again, more on this later.)
- Racial Benefits. Some races get bonuses to one stat and penalties to another. This one's short and sweet but check it out. They don't matter at all.
- Increasing General Power. Increasing your primary stat does a number of things for your character and is built in to your class every couple of levels. What do you do about that?
Character specialization is something I've been thinking a lot about recently, especially because of my recent interest in Fate. In that game, you have aspects and skills/approaches and that's it. It's not really necessary to know exactly how strong your character is in terms of weight carried because it's not important to the narrative. Conan never worries about the weight of his gear because he can carry as much as makes sense and the only time anybody cares is when he's being chased and- oh no- he can't carry the gem of Clizzak-teff and escape the lizard people! Will he fight or stay?
D&D has a long history of sort of doing this backwards. Nothing in D&D says "this system will produce the interesting stories you want to tell." On the contrary, it's "this is the story you will be telling; make it interesting," and as long as you're interested in a story where you're on a dungeon-crawling treadmill and counting your coins, then you're set. 
Where am I going with this?
My point is that attributes don't really impact the story. When you're asking for a dice roll, what you're asking is not "are you strong enough to do this" or whatever, you're asking "are you capable of this?" Nearly every roll that one would use an attribute roll is already covered by another skill. Brute force is a subset of Athletics. Dodging is Acrobatics. Knowing history is History. If you can think of a use for an attribute that isn't covered by a skill, I am extremely interested in it. I can't think of any.
See, here's where it gets a little fun. The math's not right without the attributes being rolled in with stats somehow. You'll be a little short on most rolls (and a little ahead on some of them.) So what do you do? That's actually surprisingly easy.
Give everybody a flat, unchanging +2. Call it a "Hero Bonus" or something, and apply it to anything that cares about any attribute bonus. Add it to your attacks. Add it to your AC. Slap it on all of your skills. When you're proficient in a thing, add your proficiency bonus on top of it.
Yeah, that means that everybody's sort of got the same capabilities, but who cares? Was that -1 Intelligence on your barbarian important to you mechanically, or do you ignore it half the time anyways in favor of playing your barbarian the way you want? Was the -1 strength on your sorcerer the only thing keeping you from running in with your dagger and getting some quick stabs in or was that always a stupid idea because you are more or less made of fireballs and are wearing a tattered nightgown? 
A bit sticky are attribute saving throws. How do you mark down an attribute saving throw without attributes? Well that one's easy: You don't. Attribute saving throws literally don't do anything at all now. Every save works the same way: Hero Bonus + Proficiency Bonus. Hey look, we've got a unified saving throw again! Isn't that neat? I'm getting whiffs of Swords and Wizardry here.
If you don't like that, then keep attribute saving throws, but rename them. Your characters are now Strong and Dextrous. When you have an attribute saving throw, then you get your Proficiency Bonus, otherwise it's just your Hero Bonus. Look at us! Saves space and only comes up when the game asks for an attribute saving throw, specifically, to resist a thing.
Here's a really neat thing that you can do, now that you've got +2 in everything- multiclass! If you ever wanted to be a Barbarian/Sorcerer you're in luck. I'll admit that it's not perfect, since the one thing that 5e really does well is give interesting subclasses and you can kind of do that already but hey, who cares? You can go Barbarian 1/Sorcerer 9 and won't have wonky stats that make you worse at both. You can be a Wizard 2/Cleric 2 without feeling like a dork because you're only getting part of the benefits.
You know what else is cool? There are no more optimal and suboptimal races. (Well there still are but it's less so, now.) Every race is equally good at everything else, while still feeling unique since they've all got good racial skills. Dwarves are still stout, elves are still graceful, half orcs are still violent sociopaths, much less changes here than you might think. Elf barbarians are on equal mechanical footing with half-orc barbarians, and dwarf rangers are as good as half-elven ones. Neat, right? 
I do recommend, if you're using this, to use one of the variant Human traits, because the human racial of "+1 to everything" was always pretty lame but it's extra lame when that's "+1 to every nothing." I personally like the skills one best, but do your thing. You also might want to change up the Mountain Dwarf- nobody's going to be a mountain-dwarf fighter if all they're getting is armor feats they had already, so something as simple as "if you already have this, then grab another feat" or make something new up. But what's a rules hack if you're not changing things? And such small things they are!
The last thing I can think about is the fact that attribute bonuses are a form of progression within the game. You're expected to eventually reach the end limit of 20 in your attribute at some point, and you'll probably leave the other attributes down in the dust. This one's tricky, because part of me is saying it doesn't really matter, while another part of me is saying that the DC of casters' spells relies on their attribute bonus, at the very least. You're really going to want to use the Feat variant that replaces attribute bonuses for characters, and you might want to add a "Spell Mastery" feat that increases the DC of spells cast by that character by 2. It's nice and narrow and means that the character has an emulated Intelligence bonus of +4 solely for keeping up with any mean NPCs who have managed to increase their spell defending abilities somehow.
Let me reiterate how this hack works in nice and easy steps for any interested tinkerers.
- Yank out attributes.
- Include a +2 Hero Bonus. Apply this bonus to literally everything that requires an attribute, including hit points, attack rolls, skill rolls, and saving throws.
- Decide whether to include attribute saving throws.
- If you do, write the adjective version (i.e. Strong, Tough) and apply the proficiency bonus to those saves.
- If you don't, decide right now whether to give the proficiency bonus to all of the saves or none. This makes heroes either slightly more durable or slightly more vulnerable to save-based attacks.
- Use a variant Human trait and consider the Mountain Dwarf.
- Use the "Feats instead of Attribute Bonus" variant and consider adding a "Spell Mastery" feat that increases spell DCs by 2 for any spell that character casts.
- Bask in the glow of a slightly shorter character sheet and slightly more even characters because you're done.
: I actually really, really like this level of play and wish D&D would figure out what it wants to be. I like coin-counting, expeditionary dungeon-delves where I'm trying to figure out how many rations I need and if I'm going to need to get an ox-cart to haul supplies and setting up base camps to retreat to at night. I like spiking doors and counting the hours left on my oil supply. What I don't like is when D&D forgets what it is and pretends it's something it's not. 5e does a pretty good job, actually, barring a few odd examples like the Outlander's complete removal of any need to forage, and some of the variant rules in the DMG that don't make any sense. It's certainly doing better than the last two editions, at any rate.
 What I'm trying to say here is that your class is self-defining and attributes don't do anything about that. Wizards are almost unarmed but cast spells. Paladins wear thick armor and smite their foes. Slight differences in attributes mean very little and edge cases, like a gnome ranger with 8str and 10 dex, are signs that either its creator is attempting a joke at the expense of their fellow players, that the rules of the system are not understood, or that they're attempting something extremely unorthodox intentionally as some sort of challenge or statement. None of these goals are harmed by what boils down to a standardization of attributes, unless I am misunderstanding something.
 This I expect to be a matter of personal preference, since one of the reasons that people like D&D is that it gives them something to build around by handing them problems to solve, i.e. "how do I make this gnome barbarian good" and all that. That's fine, but I feel like 5e isn't the right game for that anyways. The classes are all straightforward and the feats are compartmentalized in such a way that I have a hard time envisioning alternate and unusual ways to make something work. Instead, 5e tries its hardest to tell you "high wisdom is just plain better for clerics" and hammers it home by having half of your class abilities and all of your spells working off it and then limiting your attributes anyways so the only real benefit of having a high wisdom race is being able to hit that magical number 20 faster and then you have to improve else anyways, or else grab a feat (if you can.) I really like this. Some, surely, do not.