The Guts Rule. With Strength of 18 or higher, a Medium character may ignore the two-handed property of a melee weapon. A Small character may ignore the Heavy property of weapons.
Effect. Makes Fighters into even more of a blender, interaction with the Dual Wielder feat will pile on extra damage. Realm of believability stretched when characters swing around oversized weapons efforlessly. Possibly the most unbalanced houserule.
Learned. For each positive Ability Score bonus in Intelligence that you have, you may take an additional Tool, Instrument, Vehicle, or Language proficiency.
Effect. Makes wizards a tiny bit more versatile. Gives intelligence a bit more use apart from skill rolls.
In Training. While training with a language or a tool, you may add half your proficiency bonus to rolls made with that tool, and communicate basic ideas when speaking a language you're still learning.
Effect. Makes it a little more natural to show that characters are slowly growing in areas other than experience points and levels. Might be abusable so you should probably limit it to one thing being trained at a time.
Wear Them Down. If you miss with a melee attack, you still do damage equal to your Proficiency Bonus.
Effect. Makes the game slightly more like 13th Age, gives martial characters a tiny bit more power as they can still deal 'chip' damage.