My first instinct is to suggest improvements to Heavy Armor.
As it is, Heavy Armor comes with a lot of damning qualities. Not only can it lower your movement speed, but it also puts you at disadvantage for stealth, takes forever to put on / take off, and can impose penalties during rests (Unless your DM allows you to sleep in armor, you better hope any ambushes come during your turn to keep watch!). To add insult to injury, characters with high Dexterity can earn AC scores that rival all but the 18AC Plate without any of these set-backs.
Yes, a 20DEX Rogue is supposed to be nimble and difficult to hit, but it is bordering on the absurd to imagine that his AC is on par with a Paladin or Fighter wearing a veritable fortress of steel. To further illustrate this discrepancy: Heavy Armor is what a professional soldier would wear to battle; Studded Leather is what a Rogue wears when he's pulling a heist... To put it bluntly, it's not fair. I tend to think that armor, especially Heavy Armor, should provide some level of native damage reduction. Sure, that attack roll of 19 is going to hit, but I imagine the damage sustained will vary depending on whether you're dressed for stealth or for war.
With the current system, Heavy Armor is pretty much relegated to either the clumsiest of PCs (Dump Stat: 8 DEX) or the Variant Human who picked up Heavy Armor Mastery at level 1. Nearly every other character is going to favor Light & Medium Armors, especially if they can rely on finesse weapons to deal damage. Honestly, it seems like the only people who choose STR are Great Weapon Masters, Shield Masters, and the rare Grappler. Everyone else can run their weapon attacks through DEX, and are thus incentivized to treat STR as a dump stat. If Heavy Armor were a more attractive option, you might see more characters investing in STR as opposed to defaulting to the "mega stat."
However, your issue was never about Heavy Armor being underrepresented… It was about the Strength stat being less useful than Dexterity. While improvements to the way AC and damage are calculated might make Heavy Armor more appealing, it does nothing for the STR archetypes that have nothing to do with Heavy Armor (the Barbarian, the Pirate, the Thug, etc.). This brings up a potentially divisive solution: stat requirements.
Speed or stealth based characters (and absolutely casters) are often guilty of using Strength as a dump-stat… but what if their weapon selection was restricted based on their STR and DEX scores? For example: If your Rogue wants to use a Rapier, she needs to qualify with 11 STR & 13 DEX prerequisites. Otherwise, she'll be better served using a light weapon like the Scimitar or Dagger, which has no STR requirement at all. This "prerequisite system" could function the other way, too. Perhaps your Great Weapon Fighter wants to make use of a new, +1 Great Sword; however, his 8 DEX prevents him from wielding anything more complicated than a Great Maul. You get the idea.
But honestly, the issue with a "solution" like this is how extremely annoying it would be for the players at the table. Just figuring out which weapons you are capable of using would be a chore, and adding so many numbers would shatter that simplistic feel that 5e strives for. A prerequisite system definitely helps to incentivize a more even STR and DEX distribution, but at what cost? Not to mention, with ASIs being so rare in 5e, you'd have to make some very tough decisions: "Do I boost my STR for more a higher STRmod? Or do I improve my DEX so I can wield a better weapon?"