The I can't fix everything rule - I want my players to play as the characters they want. There are lots of illusionary choices in 5e character design, places where you're presented with a host of options but really only a couple are right and all the rest are wrong. I'm just one person and not a team of designers, and can't rewrite the entire book, but if you want to play something that is discouraged by the rules, like a two weapon fighter, talk to me and we'll figure out how to make the 5e system conform to your vision.
Holding Breath - I had an encounter go badly due to the ambiguity and absurdity of the Holding Breath rules, so I tightened them up.
In initiative, the time you can hold your breath is mapped to rounds instead of minutes. You can't fight to the death or run a half mile while holding your breath. 2-4 rounds is much more reasonable than 2-4 minutes and combat could actually last long enough for it to matter.
If you are suffocating, most offensive actions are at disadvantage, but defensive ones are not, to encourage characters to find air instead of continuing to fight.
Dodge - In the 5e race to 0 hp, Dodge cannot help you win, it can only make the race take longer. Some classes really only have one action they can take - I attack!
I buffed Dodge by allowing the character to perform a minor action - the kinds of things players often say "can I ___ as a bonus action?" It's not a bonus action in this place, it's part of Dodge.
Taunt to encourage an opponent to attack you
Threaten to try and scare them off
Feign to get an Opportunity Attack if attacked
Bulwark to gain ProficiencyMod damage reduction from one damage type (B/S/P are three separate types)
Observe to get some insight into a more effective way to defeat the opponent
Focus to calculate Concentration checks by Damage/3 instead of Damage/2
Cantrip to create a space for the host of 5e cantrips that are rarely used because they're not worth an action in combat, like True Strike and Blade Ward. You have to target yourself or an adjacent space in this case.
My players are much more likely to Dodge when combat isn't going well, but combat hasn't become a Dodge fest, so I'm happy with it. It probably steps on Battlemaster's toes, but a) I don't have anyone playing a Battlemaster, and b) I'm in the camp that most of the things a Battlemaster does should be available to anyone in some way. It also is an indirect buff to Monk, since they can Attack and Dodge at the same time.
Dying - Dying in 5e feels like you have a clock floating above you that says how long your friends can ignore your plight, instead of the tense moment it should be. Also, crit fumble mechanics usually suck, and this is the only place in 5e where a nat 1 is worse than a regular failure.
When you reach 0 hp, you do not fall unconscious, you gain the Injured condition, which includes all the RAW stuff about rolling each turn and what happens when you get hit, but also gives Disadvantage on most offensive actions.
The Nat 1 double failure is gone
You die when your Death Save Failures reach twice your Proficiency Modifier
You stop making Death Save rolls if you gain any HP or the Stable condition (3 successes, Spare the Dying, Healer's Kit, etc), but if you're still at 0 hp you still have disadvantage on offensive options, and attacks against you still inflict Failures.
Death Save Failures accumulate, and are only removed by healing to full HP, or by a few effects like Restoration.
I have plenty of other minor tweaks to individual classes, weapons, spells, etc., but a lot of them have already been covered here. One I'd really like to implement (next campaign!) is overhauling Constitution to provide more HP up front, less at later levels, and more side benefits, so everyone doesn't take 14+ CON. No one would play Raistlin Majere in 5e.