One of my house rules involves cleric weapon proficiency: each of the gods in my setting has a favored weapon, which clerics are treated as proficient in, whether they normally would be due to race/class proficiencies or not. Not all clerics will make use of that proficiency necessarily, but it does make them stand out a bit and you can identify at a glance "oh that cleric with the war pick worships the Ebon Judge."
In thinking about it, I guess I do this, too. It isn't an issue with the current PC Cleric, but I've changed weapons on NPC Clerics to fit with their gods and wouldn't bat an eye if a PC Cleric chose a god with a particular favored weapon. I really like Clerics to not be samey-samey.
Otherwise, we're generally making an intentional effort to play by RAW. We "sat out" most of 4E and came back with LMoP to see how things had changed. LMoP was explicitly RAW and settingless. When we finished it, I gave the players the option of me creating a new home brew setting or using Eberron. Eberron won out, mainly because we're in a beer-and-pretzels mode. As others have said, 5E doesn't really
need a lot in the way of house rules. The few we do have are generally to fit the setting:
* Eberron races are available and per the UA article. Other published races (genasi, goliath, etc.) are available, but generally rare and subject to strange looks even in Eberron. No drow. All starting PCs are still in play, so they're all from the non-optional PHB.
* All PCs start with one feat. This is either a conversion of 3E action points (which was explicit in Eberron) or a Dragonmark. I created a feat for this before the UA article went up. They're similar, but mine has more options, so that's what we use.
* The second UA Ranger (with 2d6 hit dice) is in use, but we nerfed Ambuscade some. The most recent Ranger is also available.
* The Mystic is available for play, but no one has done so.
* I converted the Sorcerer dragon bloodline to a Warlock pact to make my daughter's first character easier to play. It works well, FWIW, though the bonus to hit points, combined with a high Constitution and the Tough feat makes her substantially more durable than anyone else in the party, including the elven (dex-based) Battle Master.
* I bought several distinctive d20s -- one for each player. These are used to track Inspiration. When you use Inspiration, you must roll that special die for the bonus die, then turn it in to the DM (me). As a bonus, you can use Inspiration to create disadvantage when an NPC has to roll a save to resist an effect you create or when an NPC makes an attack roll against you (both declared in advance). I'm not sure why this is almost never used, but a couple of players have figured out that, because the player rolls the Inspiration die, I did that to explicitly give them a chance to gauge the strength of major foes and/or to verify the roll can't be fudged (not that I typically do that, but appearance matters).