I think there has already been some discussion of this in some of the houserules in the homebrew forum
For me, the problem with int is that the skills it provide seem nebulous. Charisma is in a similar situation as a supporting stat, but I often find myself looking at charisma builds due purely to the skills.
Charisma has three main skills. They each contribute to social situations (filling a complete "Face" concept), and they have a variety of in combat uses. The combat uses usually provide a solid little benefit that can be obtained at level 1 without feat investments (although intimidation in particular gets some really nice feats- both skill and class), and they work well as the "third action" that doesn't affect map.
In comparison, int has a larger number of skills, but they seem unrelated. Crafing, society, and knowledge skills seem hard to put into the same build without being a generic "I got a bunch of random skills" build. And each of these are hard to apply in combat without taking highly specific feats.
And even with the suggested skill inscreases, I don't think I would go for int. As a martial character, you are probably only going to support one mental stat unless you are doing for a pure dex build. And The int skills don't feel like something to invest heavily in unless you are doing a knowledge build, since you would lack the skill feats to support all of that.
Now, rogues like going all dex, and they have the skill feats to support all those skills. But... they also make the suggested house rule moot, since they get a ton more skill increases from their class alone.
If you want to held int with house rules, then it is easier if you do changes so int can work for more skills. The knowledge skills seem like an obvious target, since the int/wisdom divide is one of the biggest factors holding them back. Lore bards and monster hunter ranger build their entire characters around avoiding that divide.
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My suggestion.
Level 3: if your Int is at least 12, gain an extra skill feat or skill increase.
Level 7: if your Int is at least 14, gain an extra skill feat or skill increase.
Level 11: if your Int is at least 16, gain an extra skill feat or skill increase.
Level 15: if your Int is at least 18, gain an extra skill feat or skill increase.
Level 19: if your Int is at least 20, gain an extra skill feat or skill increase.
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I love how to some there's a "general consensus that int is a poor secondary stat" and to my group extra trained skills and languages is valuable.
I think this general consensus is exaggerated, and based on the assumption that general feats/ancestry feats are of such low worth that you'd happily spend them on untrained improvisation.
But if not going up to expert is a dealbreaker, then the same +2 difference between just adding your level and the +2 from actually being trained is... what?
Also, we can expect more and more general/ancestry feats over time, so the value of those slots is expected to go up.
I don't think Intelligence is really that far behind other abilities in the general case; every class has some abilities it's happy to leave at 10. Charisma and Strength get told off a lot too, while the full plate contingent leaves Dexterity at 10. Everyone would want Constitution and Wisdom high, but at 10 or 12, most characters are still viable, and sometimes your class is just MAD enough that that's what it's gonna be.
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My most successful PFS character so far has been my fighter/wizard, who's bailed out a lot of parties on skill challenges or just by knowing stuff.
It's not so much that everyone in the party has to have high Intelligence, but someone has to. Just like not everyone has to have high Charisma, but someone should be able to do face skills well.
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My own Int House Rules are as follows:
Ability Score Change:
Intelligence: When your Intelligence modifier rises to +5 or higher, you may increase your Proficiency in an existing Skill rather than receiving a new one at Trained.
New General Feat:
Skill Expertise (Feat 7, General):
Prerequisite: Int 16, Trained in at least two Skills,
You may increase two Skills from Trained to Expert. This Feat can never increase any Skill beyond Expert (though other effects work normally to increase the Skills gained this way). You may take this Feat a second time if your Intelligence score is 22 or higher.
New Skill Feat:
Expansive Knowledge (Feat 1, General, Skill):
Prerequisite: Int 16, Trained in Medicine, Nature, and Religion.
You gain a +1 circumstance bonus when making Recall Knowledge checks with Medicine, Nature, or Religion. If your Intelligence score is 20 or higher and you are a master in at least one Int-based skill, this bonus increases to +2.
They're all relatively low impact, but all also reward Int investment, which is, I think, the thing needed.
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Trained skills are great at lvl 1, but have fallen way behind later.
People say this a lot, usually close to "if you're not taking this skill to Legendary then why even bother". But that folk wisdom doesn't really stack up against the game's DC design.
If you look at the Level-based DCs in table 10-5, you see that a level 1 DC is and a level 20 DC is 40. So in 19 levels, the DC went up by 26. In other words, if all you do is become Trained, by level 20, you've become 6 points worse than you were at level 1. That sounds bad doesn't it?
But what if your ability score went up at level 5, 10, 15, 20? It might not go up at each of these times because when your stat gets to 17 and above the growth rate slows down. But the abilities that hit that barrier first, those are skills linked to your key stats and odds are you're choosing those to make Expert+. The abilities that hit that barrier later or never, those are the skills you're leaving at trained. It's a minor paradox: the skills you're worse at in the beginning age better (without effort) than the ones you started great in. I think you can expect to improve about +3 by level 20, so a Trained skill with no further attention paid to it falls behind by only 3 over 19 levels of playing.
And then there's item bonuses, Aid Another (which actually becomes reliable and stronger at higher level), and buff spells - a Trained skill isn't going to fall behind all that much, maybe 1-2 points in the end.
Yeah, if you really care about a skill you make it Expert, but if you just want to be able to do it as a secondary character or be okay in skill challenges where everyone has to roll, then Trained is fine.
I think Lore skills are being underestimated, as well as extra skills overall, though that depends a lot on class.
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For the record, I think Trained Skills are pretty useful as well, which is why all the House Rules I list above are relatively modest improvements that only really apply if you make them...but they do give people who don't think Int does enough a bit of a boost to it, which I think is good and useful.
For context, I also allow either Wis or Cha (player's choice) for Will Saves, so I think the only 'weak' stat I haven't powered up a bit is Str...and Str does just fine IME.
I love how to some there's a "general consensus that int is a poor secondary stat" and to my group extra trained skills and languages is valuable.I think this general consensus is exaggerated, and based on the assumption that general feats/ancestry feats are of such low worth that you'd happily spend them on untrained improvisation.
It may be worth noting that many archetype dedications also give a trained skill or even a skill increase, so players using the free archetype variant (or who just like to tinker with archetypes) could potentially end up with enough extra skills to make INT's base benefit more redundant.
Also, item/status bonuses and aiding one another are very helpful for catching up, but not always available, and chances are already scary for the same reasons all of the other math matters more in PF2 — so it makes sense for the common wisdom to be "stick to the best you've got", though also true that the greater variety of ways to improve these checks later on isn't immediately obvious to the Lv 5 character stressing about whether another trained skill will ever matter.