Now I am not talking here about turning HeroQuest into a full fledged RPG on the level of a CyberPunk, where you are constantly referencing tables and adding modifiers to multiple types of multi-sided dice to account for every possible thing and having character sheets that start to look like legal documents, but...

I am as usual starting with the NA/Remake rules as the base and then moving on from there...
Rather, I am talking about the kinds of things, that I probably will never use, but people talk about... it seems a lot of people get upset at some point in their HeroQuest careers because they learn about something in history or from some youtuber sword enthusiast that differs from how HeroQuest does it and wants more "realism" in their game. Usually they stop short of overhauling the whole thing, and just implement one or two little changes. What might those be?
Now obviously I am not alleging that real magic and monsters actually existed in the middle ages or that HeroQuest takes place on earth. I'm just having a bit of fun here to help people think about what everyone talks about at some point or another. Are all of these things practical? At the end of the day you may end up with a completely different, even "broken" game rather than the one you grew up enjoying (fair warning!). And yes, you'd be mixing time periods and regions here too if you really wanted to separate things out. HeroQuest is supposed to bring together characters from around the world to have their adventures not just a single culture or kingdom.
What is an "Elf"? Or a "dwarf"? Is the Dwarf a human with dwarfism or is he some kind of fictional "race" of creature like the Elf who has different abilities to you and me? How does the Wizard's magic really work? We might be back to inventing rationalizations for why the gameplay works the way it does vs. changing it to conform to what we think might be "realistic" but that stuff is more speculative. Most of this conforms to the "coat of paint" and how the hero's equipment works.
I'm going to start a list just to help get people started...
1) Terms are inaccurate. This is a controversial one because many of the terms we use for stuff from the middle ages/medieval period are not the terms anybody back then actually used. Nevertheless "that's not a ___" is a common retort. Chainmail should just be "mail" and "Plate Armor" (or Armour if you prefer) is what the other stuff should be called. That's not a broadsword. Why is the helmet open-faced like that (yes, such helmets existed).
2) Plate doesn't slow you down. Even if you found an "ill-fitting" suit of armor, generic armor did exist and there's no reason why all four heroes can't wear it with no penalty. Is it a complete suit of armor or just a breastplate? Does the armor provide full coverage or does it have gaps and things? Yes, it was possible for a knight to put on his own armor (not as easy as throwing on a sweatshirt, but quite possible without a "squire" to help him). Yes, Jousting armor was more restrictive and heavier but still no cranes required to lower you onto your horse and you weren't a helpless turtle stuck on its back if you happened to fall down.
3) Why is the Wizard limited? If he's a normal, able bodied human adult, then there's nothing stopping him from using the same gear the other heroes use. How do his spells work? Maybe there's nothing stopping his magic from being used by other heroes as well?
4) Limited Carrying Capacity. Here is that rule some people love so much. How much can a hero realistically carry and still be combat ready? You're going to have to drastically limit his money (how big are those coins? are they pure gold?), how many weapons he can carry, etc. How big and heavy are potions? Are they fragile? (glass or ceramic bottles? or maybe heavier metal or wooden jugs?)
5) Limited stock. Yes, armories may have limited stock, but probably they have enough for all the heroes to spend all their gold if they wanted to, they just can't carry it all into each quest!
6) Upkeep, room & board, taxes... the heroes will have to spend some of their money in tribute to the King or whatever, medical expenses, paying the innkeeper, blacksmith and tailor making repairs between quests, etc. They'll also need time to rest ("a few days/weeks later" does the job). They will also be making reports to their benefactor(s) and paying for travel expenses (horses, ferries, trail guides, maps, etc). Anything a medieval person might have to spend money on (or forage for) the heroes will do, but a lot of this can still be left out of gameplay for brevity's sake even if acknowledging it. Simply camping for the night and resting for a few hours before they walk through the exit door or head back down the stairs makes sense. Hiking back out to the next town to buy things also makes sense. You could also have your gear modified.
7) Haggling and barter. When they buy stuff, why does everything just have a fixed cost? Perhaps some shop keepers are willing to bargain or accept things in trade. Why only get half your gold back for a piece of equipment you never even used? This could be tinkered with quite a bit.
8) Two handed. Why does the staff require two hands? You could wield it in one but maybe it's more useful with two. Why can't you use it with a shield? How big and heavy is the shield? You probably could wield a shield with a staff or Battle Axe, actually. Or maybe the longsword actually would require two hands (for full effectiveness). What kind of crossbow is that? Doesn't it take two hands to load and two to fire? (or is it some kind of shorter range/less accurate pistol crossbow that can be fired AND loaded if you're clever, with one hand?).
9) I don't worry about crossbow limitations or range limits on dagger throwing because world record holders can achieve some pretty awesome feats with these and you're "heroes." I also don't worry about running out of arrows because you probably carry enough for what you'll need (but they WILL require gold and WILL affect your carrying capacity/weight). Don't worry about wind speed because you're indoors most of the time, right? Not only can the crossbow hit all eight squares surrounding you, but it should also be able to fire at point blank range.
10) Wounds. How are you staunching the bleeding? Magic healing can't always be the thing, so that wound is going to get infected. I hope you bought bandages and have room to carry them as well (are you tearing off strips of your clothing)? Are you going to simulate disease? (have the heroes had all their shots)
11) Loot the bodies. No reason why you couldn't other than honor or law... why would monsters not be able to use your gear just as you could use theirs?
12) Searching. The existing mechanics kind of make sense but you could change it to say that the first searcher is so thorough he finds everything and there's no need for anyone else to search. Why would seeing a monster in the distance stop you from searching? (perhaps you just search the "safe" area only). Searching for Traps is thorough too, right, so why would you still get "Hazards" and undetectable traps (are they magic??). And well of course you have to "open the chest" and move up to it to get what's inside. You probably have to touch nearly every square of the room to get what you want. Why would you not be able to search corridors ("passages") for treasure as well? Again if you can search safe areas that includes your buddies holding the bad guys back while you check each square or whatever. You can search again and again but you won't find anything because you've already stripped it bare. Is that furniture worth anything? Maybe you can chop them into firewood or sell some of those books from the bookshelf? (if you have the inkling to lug it all the way out of the quest!).
13) Death saves. As you are dying are you really able to reach into your bag, find the right potion, pull it out, uncork it, drink it and the magic works that fast to heal any injury? (It's MAGIC after all, what are its limits?). Maybe you can, or maybe you can't do this. Maybe a hollow tooth that holds that healing juice or a bag that's always in your mouth you can bite down on, or like one of those "beer hats" with the straws? If a monster stole your potions couldn't he do the same thing?
14) Darkness. Are these sealed dungeons or treasure vaults or tombs? Because how can you see in the dark? Do Elves and Dwarves have magical vision, or are you fumbling around constantly. Are you carrying a lantern? Because sources tell me that "torches" like you see in the movies are usually terrible... too bright in your eyes, smoky (cough cough even harder to see in narrow passages, soot and stuff falling everywhere and they don't last very long unless you carry many of them or supplies to make more... with limited carrying capacity of course). Then again there are longer torches that last longer but are even harder to carry (maybe require two hands). Are you dropping the torch and pulling out your weapon every encounter? What are those lanterns burning? Are they fragile or a fire hazard with dry materials in those dungeons (or is it mostly slimy stones and such?). Magical flashlights?
15) Navigation? How do the heroes know where they are going? Does their compass work in the dark or do they have maps? We players have a 360 degree God's eye view of the dungeon (as does Zargon) but do they? Do heroes have to shout or blow whistles to keep in contact if they split the party? How do they know where everyone is (magical walky talkies?). Shouldn't monsters have the same limitations or do they have a hive mind or something? Are there communication tubes like on a ship where monsters would have to stop and ask for orders?
16) What are those monsters doing before they're discovered? Also, they should be able to move about the dungeon and open doors, move furniture, move treasure, re-stock weapons, re-group, etc, right? Or are they magical guardians that only come to life when the heroes open the door? Can monsters hide inside cupboards or spring out of pit traps? And yes, you should be able to dig through sprung falling rock traps (did you bring the right supplies?) or fill in pit traps or brace them, etc. (assuming you had room in your pack/on your back for that stuff!).
17) Movement. Why is movement limited to 2d6? Shouldn't heroes just be able to keep walking and walking (or running for short distances) if they want? Is it because of their heavy packs? Can they set those packs down and move faster, then go pick up their gear again? Maybe their combat movement is different from their "everyday" movement? Do heroes get tired and need to rest?
18) Food & Water. How long are these quests? Other than needing to rest or even sleep, is there room in their pack for provisions? Is there anything to eat in those dungeons? Are they eating the rats and spiders raw or did they bring supplies to make a fire and cook them? Maybe they feasted before they got there, but at least they should have a canteen. Potion of Hydration??
19) Bathroom break. Hey, how big are their bladders on this trip? Are they vulnerable when taking a "pit stop"? Do the monsters have a privy?
I'm probably missing some big ones, so please let me know what else you think of.... again, not trying to design a new system I plan to use here, just mostly brainstorming because it's one of those things that come up so often...