Everyone likes different kinds of DnD. For me, my favourite experiences have been roleplay intensive, testing the boundries of the world, and involve just the right amount of silliness that means everyone has a good time, without going overboard and destroying the credibility of the world we're playing in. When I DM, I try to include different aspects of DnD to cater to all my players' preferences, but since it's imperative that I'm having fun as well, a lot of my homebrew rules tend to lean towards this kind of roleplay heavy DnD. What works in your group may not be the same as what works in mine, but here are my 5 favourite homebrew rules which I include at my table.
Dreams & Nightmares
I don't know about you, but I sometimes find night-time can be a bit boring in DnD. Your party goes to sleep, perhaps they share watch between them, maybe roll some perception checks to see if they notice anything weird going on around them, and maybe you've got an interesting night-time encounter planned. That's all good, but a lot of the time it's literally just 'you don't notice anything, the night passes uneventfully and everyone wakes up with full HP and all spell slots restored'. A bit of a snooze (pun noticed but not intended) if we're honest. Enter my dreams and nightmares mechanic.
Full disclosure, this is actually fairly prep intensive - but I find it adds some really awesome moments to my campaigns. Basically, each time the group takes a long rest, they each have to make a constitution saving throw (you can add modifiers due to the expense of the accommodation they're springing for, or any things your players come up with to try to improve their nights' sleep), and then the member of the group who rolls highest will get a good dream, while the player who rolls lowest will get a nightmare. I have these written out in my prep (a table with 1 good dream and 1 nightmare for every player, which I will then replenish whenever one of them gets used). If your player has written a good backstory, you should have lots of interesting plot threads to weave into these dreams or nightmares, in weird and wonderful ways that you couldn't explore within normal waking hours in game.
I find the nightmares particularly enjoyable to write, as I can play with their characters' fears in really strange, and unsettling ways. After recounting their dreams and nightmares, I then have another d6 table of good and bad outcomes depending, which these players can roll for. A good dream may result in you getting a d6 of temporary hit points or a bonus cantrip for 24 hours, for example - while a bad dream may give you a level of exhaustion or to lose a spell slot etc. I find this added dimension to night-time adds great roleplaying opportunities, a chance for players to go deeper into their characters psyches, and another level of jeopardy. If you don't mind the additional prep time (which isn't too bad after you've made the initial table of dreams and nightmares, and then only have to replace a couple at a time) then this really can add a lot to your game!
Character Rumours
This rule is specifically designed to try and get my players chatting to each other more in character. When getting them to come up with their character ideas, I also ask them for two rumours about their character - 1 true and 1 false. I ask for these in advance of session 1, so I can give them out to the other players (in such a way that each character receives rumours about 2 other players, but they don't know whether they're true or false).
Once their characters are introduced then, it automatically gives them things to talk about, or things to try and deduce about the other characters as the adventure unfolds. If you want to expand this idea, you could also give your players ruomours about things, places or other people in the world, or indeed you could have some of the NPCs in the world having heard the rumours about your characters. It's up to you how far you want to take this, but the principle is that it both gives your players reason to roleplay with each other, and hopefully increase the depth of the world and your players engagement with it.
Shared Backstories
Again, in an attempt to get my players to roleplay more, and to give them reasons to come together as a group in the first place, I ask their characters to have some degree of shared backstory. How extensive this is, I leave completely up to the players - they could be really close like partners or family members, or it could be something really trivial, like they once met each other in a tavern. Whatever it is doesn't really matter - so long as it gives something for the players to talk about in character.
For simplicity's sake, I don't ask for every single player to have a shared backstory with everyone else. Instead, before we start the campaign and while they're still coming up with their character ideas, I split the party into groups of 2 or 3, and ask them to have a backstory between them. This, combined with the rumours mentioned above, means that right from the outset, the party have reasons to talk to each other and can begin to think a bit more about both their own and the other character's backstories.
Changes to Death Saves
To try and increase the tension of death saves, there's a couple of extra mechanics I introduce during this scenario in my games. The first I stole from a reddit post I saw and loved, which was to give my players the option of a 'heroic sacrifice'. In short, at any time a character has to roll a death save, they can instead invoke a heroic sacrifice. That gives them the chance to immediately stand up and have one last turn. At this point, all spell slots and abilities are refreshed, all attacks made at advantage and all hits are critical. You can bend the rules however you see fit here to make the character's final act awesome. At the end of it, however, the character dies immediately, and cannot be revived in any way. I love this rule as it gives players an epic way to end their character's journey (perhaps in defeating their arch-nemesis, or saving another party member). You can then work with the player to create a new character that fits the setting and can be introduced to the party next time.
If a player doesn't wish to say goodbye to their character though and wants to try their luck with death saves, I have an additional mechanic to try to increase the tension and get all players more invested in the character's fate. At this stage I will give a bonus (say+1 to the death save roll) if the character also recounts a 'life flashing before their eyes' memory before rolling. There's something about hearing a memory of the character's childhood, or a lost love, or crushed dreams, that increases the depth of feeling that all players (and indeed the DM) have for that character. Whether the character then ultimately goes on to live or die, the feelings of either relief or despair should be all the more intense for the player and the rest of the party. That intensity is, ultimately, what you're trying to create.
Plot Twists
These are a fairly new addition to my set up, and are born out of a desire to make things more unpredictable for me as a DM. Each time my party levels up, as well as gaining increased HP and new abilities etc, I also get my players to pick from a deck of Plot Twist cards that me and a friend of mine created. There's about 50 of these - which might involve a lock unexpectedly opening, someone discovering an unknown elixir in the vicinity, your enemy turning out to be a long-lost acquaintance, or indeed any number of different effects. My players pick these at random, without telling me what they are, and can use them at any point in game. This can lead to some brilliant moments and truly unexpected consequences. I love these as they not only give my players a little bit more agency in deciding the direction the plot takes, but they also keep me on my toes and challenge my improv skills.
Bonus Rule
Finally, anyone who comes in costume gets inspiration. Self-explanatory, but costumes are great for setting the mood and getting people in character - so anything that incentivises that is a win in my book.