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Ability scores worsen 5e. They encourage cookie-cutter point-buy characters and they slow the game down because you're calculating derived values like saving throw DCs and attack values. I don't know about you, but I'd rather spend less time mathing around and more time playing around.
Moreover, there's the disparate impact ability scores have on the game: Dexterity influences Armor Class, melee and ranged attacks, and the all-important ability to move around without alerting the whole dungeon on your head.
Let's fix that by eliminating saving throws. Instead, everything is based upon your proficiency bonus.
TL;DR Summary
There's a more detailed section beyond this point, but if you're just here for the rules, I've attached them below.
Mathing It Out
Let's make the following assumptions for a point-buy game (which is the widely-followed method of character generation in 5e).
Characters will have a 16-17 (+3) in the ability score associated with their main schtick at level 1 and a 20 (+5) at level 20.
Characters will have proficiency in their main attacks, skills, saving throws, and save DCs (+2 at level 1, +6 at level 20).
Thus, most characters will grow from a +5 bonus to a +11 bonus over the course of the game for their proficiencies.
Characters with skill expertise will instead grow from +7 to +17 over the course of the game.
Fighters will have a minimum of 16 Strength and Athletics proficiency and Constitution saving throw proficiency, wizards will have a minimum of 16 Intelligence with Arcana proficiency, and so forth. This assumption isn't an off-the-rails spherical cow situation,
Keying Everything Off Proficiency Bonus
Right off the bat, we see that removing ability scores puts expected bonus values at a -3 to -5 deficit. How to make up that math? Simple: everything is keyed off of proficiency bonus.
Attacks: 1d20 + twice your proficiency bonus.
Save DCs: 8 + twice your proficiency bonus.
Skill or Tool Proficiency: 1d20 + twice your proficiency bonus.
Skill Expertise: 1d20 + thrice your proficiency bonus.
Saving Throws: 1d20 + twice your proficiency bonus. (A note on this further below.)
This starts characters off with a +4 bonus in their combat functions and grows to a +12--a minor +1 discrepancy between the "real" 5e math, which is mirrored for skills and save DCs.
Saving Throws
Saving throws are the trickiest part of this because 5e opts to make saving throws categories based on ability scores. Ironically, this is the area of the game that is weakest, because the math falls apart at higher levels, especially with save-or-suck spells.
You will need to write down the saves and associated ability score, unfortunately:
Saving Throw Proficiency: 1d20 + twice your proficiency bonus.
But wait, you what about Armor Class, Hit Points, and class features that rely on those ability scores?
Armor Class
Armor Class needs be changed slightly. Unfortunately, with how 5e is designed, this does mean some (relatively painless) calculations.
Light Armor: add proficiency bonus to the Armor Class.
Medium Armor: add half proficiency bonus to the Armor Class.
Heavy Armor: no changes.
At the highest echelons, this will equalize light, medium, and heavy armor, but I don't think it particularly matters - the difference is a single point of AC.
Hit Points
Hit Points are an easy matter: all characters get maximum Hit Points at each level.
Compare a fighter progression of Hit Points:
Normal Level 1: 10 + Constitution modifier HP (let's say 12).
Variant Level 1: 10 HP.
Normal Level 2: 17 HP (12 + 1d10 + 2).
Variant Level 2: 20 HP.
At higher levels, the fighter is likely to have increase his Constitution, so let's push him to a 16 (this is lowballing it most likely).
Normal Level 10: 94 HP (13 + 9d10 + 27).
Variant Level 10: 100 HP.
Normal Level 20: 184 HP (13 + 19d10 + 57).
Variant Level 20: 200 HP.
All the values are pretty close. Even if the fighter brings his Constitution to 20, he has 224 HP. I don't see this particular discrepancy as a huge deal. The downside, of course, is that low-level characters start out marginally squishier. If that's a bother, start them at level 2-3 or give them a +5 HP booster shot.
Other Ability Score Derivations
Some other parts of the game rely on ability scores to determine uses or bonuses. Add your proficiency bonus to these values instead of the ability score modifier. Chances are, it'll start out a bit weaker but end stronger--and I don't think 5e is so carefully tuned that a + here or there is going to shatter anything.
Variants of the Variant
Two rules changes I would suggest in general as a modification to the standard rules:
Saving Throw Non-Proficiency (Optional): 1d20 + your proficiency bonus.
Skill or Tool (Non-Proficiency) (Optional): 1d20 + your proficiency bonus.
These two suggestions are there to "flatten" the math somewhat, as characters without appropriate proficiencies will fall behind their compatriots. Since many characters will have positive ability score modifiers that would normally apply to ability checks, this provides a buffer for them as they level.
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