Most PCs don't care much about intelligence, often leaving it at the single digits. And this makes sense, because it's frankly useless for anyone that's not an intelligence caster. So I want to know how you would make it less useless, hopefully without punishing those without intelligence too much. I currently am thinking about using the following rule: You gain one extra skill proficiency if your INT is 14 or higher. You gain one extra tool/vehicle proficiency at 12 and 16 INT.
Use intelligence checks, in particular, the Arcana, History, Investigation, Nature, and Religion skills.
From the Ability Scores and Modifiers section of the basic rules, under Intelligence:
An Intelligence check comes into play when you need to draw on logic, education, memory, or deductive reasoning. The Arcana, History, Investigation, Nature, and Religion skills reflect aptitude in certain kinds of Intelligence checks.
As guildsbounty suggested in the comments:
Don't tell your players what they are fighting. Describe the monster to them. Make them do a reflexive knowledge check to see if they recognize it (unless they have recognized it previously, or it is an extremely common creature, like a goblin). The results of their check details how much info you give them about the creature. Before a social encounter, call for a History Check to see if the bard can remember the customs of this particular land so he doesn't offend anyone...or at least comports himself in a respectable way (possible advantage on success?), etc.
There are other uses for intelligence. For instance, on the Astral Plane, intelligence determines how fast you travel. From the DMG, p. 46:
A traveler in the Astral Plane can move by simply thinking about moving, but distance has little meaning. In combat, though, a creature's walking.speed (in feet) is equal to 3 x its Intelligence score. The smarter a creature is, the easier it can control its movement by act of will.
There is one official variant rule that may help a PC who already has a good intelligence score feel like it is less wasted:
Normally, your proficiency in a skill applies only to a specific kind of ability check. Proficiency in Athletics, for example, usually applies to Strength checks. In some situations, though, your proficiency might reasonably apply to a different kind of check. In such cases, the GM might ask for a check using an unusual combination of ability and skill, or you might ask your GM if you can apply a proficiency to a different check. For example, if you have to swim from an offshore island to the mainland, your GM might call for a Constitution check to see if you have the stamina to make it that far. In this case, your GM might allow you to apply your proficiency in Athletics and ask for a Constitution (Athletics) check. So if you're proficient in Athletics, you apply your proficiency bonus to the Constitution check just as you would normally do for a Strength (Athletics) check. Similarly, when your half-‐‑orc barbarian uses a display of raw strength to intimidate an enemy, your GM might ask for a Strength (Intimidation) check, even though Intimidation is normally associated with Charisma.
Justifying the use of Intelligence to bolster other skill checks is often easier than other attributes. For example, an intelligent character make an Intelligence (Persuasion) check to convince a merchant to offer his wares to famous adventurers at a discount, given the effects of their notoriety on his revenue.
That said, I'm not sure it accomplishes the goal you want it to - making PCs regret dumping Intelligence to an 8 as much as Dexterity 8 or Constitution 8. In my campaigns, I personally use the following house rule.
Each character has a number of “Lores” equal to 1 + their Intelligence modifier. A Lore is a narrow area knowledge (much smaller than a skill) that a character has from their adventuring experience or background. When a character must make a ability check related to the knowledge they have from their Lore, they may add their proficiency bonus an additional time to the roll. Examples: Contemporary politics, architecture, geology, siege weaponry, law.
This makes an Intelligence 8 character lose something, in the same way they do with Initiative for Dexterity, HP for Constitution, or carrying capacity for Strength, while not affecting balance as much as a full proficiency.
Edit: Results in my home game: players seem to enjoy have things their characters an experts in. It gives each character a narrow opportunity to step in and be the party expert - and players seemed more reluctant to pick Int 8 because it would mean they wouldn't get that. That said, it does increase the amount of "Can I apply my Sailing lore to my acrobatics roll to not fall in the water?" kinds of questions I field. If my players were less mature, that might be an issue.