I'm unsure about adding a damage bonus to all weapon styles (2h, sword+board, archery) except dual wielding. Because you need your bonus action to make your offhand attack. Where this leaves us is the weakest (after Extra Attack) weapon style gets nothing more, and all of the other, already better styles get another thing that could stack.
That's not an inherent problem with this - I'd just want to include a feat that boosts up dual wielding as well. I guess you could also solve this by allowing the bonus action for this feat also allow an offhand attack. (Though then weapon & shield users could do a Improvised Weapon attack with the shield.)
I was trying to keep the complexity down, since usually the half feats are fairly simple. Dual wielding has so many problems with action cost that I don't know of a simple way to add damage to it without making the feat passive. The whole feat was inspired by the Kensei's Shot ability, and the samurai fighter archetype, where you use your bonus action to focus and get a benefit if you hit, but waste your bonus action if you miss. I'm hesitant to put too much power into a half feat like this, even if it isn't real.
Focused
+1 Strength or Dexterity
You can use a bonus action to cause your next attack with a melee or ranged weapon cause an extra 1d4 damage. If you have the Extra Attack class feature add an extra 1d4 damage for each additional attack you would be able to make. This is added to the first attempted attack, not to each attack this turn.
This is the final version, it's not perfect, but i'd rather not hijack the thread for 1 feat that I will never actually use. I added scaling with extra attack, but it's still all or nothing attached to the next attack that turn.
To get the thread rolling again, here's another feat.
Master of Heights
+1 Strength or Dexterity
You have advantage on checks and saves made to avoid falling, such as on a narrow bridge or slippery surface.
Gain proficiency in either Acrobatics or Athletics.
Inspired by a party of 1st level wizards in the sewer, who had 2 people get knocked out slipping off a ladder.