13th Age, an OGL game from Jonathan Tweet and Rob Heinsoo that came out before 5e but shares a lot of aesthetics with it, figured out the math for their system. It was to make weapon damage be [L]dX + ability. So a 3rd level fighter with a 1 handed weapon and an 18 Str would do 3d8+4. This is for a 10 level system instead of 5e's 20 level system, so it would be growing at half that rate if directly ported. The ability damage portion doubled at 5th and again at 8th, their tiers of play. So a 7th level fighter with a 20 Str would do 7d8+10 with a martial one-handed weapon.
(13th Age also gives more HPs, with three dice at 1st and gaining more then one at some of the advanced levels. But again, only 10 levels total.)
However, 5e has more damage adds from class features, spells and feats then 13th Age does. And those can be classified as per hit or once per turn. So any system that wants to condense to a single attack roll needs to consider those types separately to keep their relative impact.