I'm pretty happy with the 5e rules, so I haven't got many house rules:
- Hit Points: At 1st level, add your Con score to your hit points instead of your Con mod.
- Healing Surge: As an action, you can spend up to half your hit dice to regain hit points.
- Feats: If you're playing a non-human race, you can trade your +2 racial ability bonus for a feat.
- Wild Magic: A wild magic surge triggers on a d20 roll equal to or less than the level of the spell slot cast (instead of always only on a 1).
- Injuries: You can choose to take a lingering injury instead of die.
I have also been using Plot Twist Cards in place of Inspiration but I'm thinking of just switching back to the default actually. My players just aren't really using the cards when they get them, so there's not much point.
Other house rules I've considered but haven't implemented:
- Adding "radiant" and "necrotic" to the list of things the barbarian's bear resistance doesn't apply to.
- Reducing the length of a short rest to 20 minutes.
- Turning the personality traits and inspiration mechanic into more of a FATE aspects sort of thing, whereby you have a pool of inspiration points that you can spend by invoking your traits, while the DM can give you more points by compelling those same traits. I haven't worked on it in any great detail, though.
EDIT:
- Setting-specific: Raise Dead only available to 3 deities/temples/clerics of those deities. 1 Good (goddess of healing & life, cuz, ya know, Healing and Life!). 1 Neutral (goddess of death, cuz she decides, "You're dead when [I say] your fate's tapestry is done and not before!" Raise Dead through her temple her may or may not come with riders/demands being made/services to be performed upon your return to the mortal realm). 1 Evil (cuz he's an elder god with "tuh powaaah!" like that...that will bring you back to be his slave/lackey for the rest of your, now, unnatural life).
I really like this one! Very flavorful! If I ever get around to running a homebrew world, I might do something like this too.
For all of you who've said you use house rules for initiative: how are they working out for you? I'm not satisfied with the default method for initiative, as I find working out the initiative order grinds everything to a halt and can sometimes take the tension out of a pending combat. I imagine rerolling initiative - and thus have to reorder the list - every round would make that feeling even worse and thus slow the game down even more. And I'm not sure I like the idea of side initiative either, in which all of one side goes and then all of the other side (assuming they haven't been completely decimated before they even get a turn). I've tried just running freeform combats without rolling initiative but my players don't really like that. So yeah - I have yet to find a method of running turn-based combats that I actually like.