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.
Call for an Intelligence check to disarm some traps
"It doesn't matter how dextrous your rogue is, your 20 DEX won't answer the question, 'Do I cut the red wire or the blue wire?'"
This means that at least one character won't dump Intelligence. :-)
Call for an Intelligence check for navigation
Reading a map is an intelligence skill, not a wisdom skill. The same goes for creating a map.
Use Intelligence for remembering things
"As you enter the town you see a group of guards in a distinctive red and gold uniform. The uniform seems familiar but none of you know who they are." (low-INT party)
""As you enter the town you see a group of guards in a distinctive red and gold uniform. Character McCharacterFace (high INT) knows that the guards belong to the order of the eagle, and the purpose of this order is…"
Use INT for Initiative
This is a rule hack that has been around for decades. The idea is that high intelligence allows you to read a situation and make plans faster than others.
In prior additions of D&D, your Int bonus gave you additional skills or proficiencies based on the score. Yes, 1st and 2nd got additional proficiencies (no skill system) and 3rd got an additional number of skills based on the score. It kills me that many PCs don't care about Intelligence in 5e.
I like your idea. Those without a high score should lack more ability to use powers, social abilities, or basic combat knowledge. It takes some degree of knowledge to see and understand the movements of a giant spider as it does to push a sword through them. Using Int for a Fighter can grant hit bonuses after identifying a creature, or finding the dragon's weak point.
This has historically (until 4e) been a thinking game based on a character. Most RP uses their RL score INT more than the characters. One who chooses to place a low INT score should lose in a number of traps, clues, and most situations that rely on cunning and understanding. So, use INT saves and INT checks a lot more. Not every check needs to be a skill. A skill is a bonus to an Ability Roll, not a roll on its own. There is no such thing as a skill check.
If someone in the kingdom provides a landmark, make PCs roll an INT to see if they know the place. If they don't, make it that they failed what would normally be a CHA (Persuasion) check with the NPC. Don't use CHA checks often.
I also switch illusion or mind-effecting spells (like V. Mockery) from WIS saves to INT. Traps are the same. Not a Perception check but an Investigation check to find them. I only use Perception regarding finding hidden enemies and Investigation (based on INT) for finding doors, traps, and so on.
Another idea: allow Ability Check reties with INT. If the warrior fails to kick open a door with STR, the magic-user can use INT to find a weak point and allow the warrior to roll again.
I think a lot of it is role-playing. If they role-play too high of an INT, punish them by not awarding or taking away inspiration, losing gold because they forgot about a hole in their sack, or adding a negative modifier to damage because they forgot to sharpen their blade.