Let me second the bloodied condition as something to bring to 6e from 4e. I think it could even be made backwards compatible with 5e: a dungeon master can give a creature that has suffered damage the bloodied condition. At that point the dungeon master can pick from or roll for an effect on the bloodied table for the creature's type or the dungeon master can use the bloodied condition effect contained in the creature's stat block. [With the tables, all monsters and NPC's might not have individual effects, and if you had a 5e monster that didn't {yet} have a 6e stat block, you could give it the bloodied condition.]
Hmmm...this leads down some interesting paths.
In 4e a creature's bloodied condition was what it was, no variance. But here you're suggesting that each creature could have a variable range of possible bloodied conditions (which could include none at all) - cool!
And to take it a step further, there could be degrees of bloodied e.g. condition x kicks in when the creature is first damaged, condition y begins when the creature hits half h.p., condition z arises when the creature has less than 10% of its h.p. left - that sort of thing.
Complicated as hell, of course, and the space it'd all take up would make the MM bigger than the full-size Oxford Dictionary - but maybe worth a look as a per-campaign system for DMs who are so inclined.
I would let the players give the PC the bloodied condition if the PC suffered half of their hp's in damage (or less if they use an inspiration point) with an effect based on class and level (multiclass PC's use their total level and pick one of their classes). I am not sure if this would work out, but I see each class having a 20 effect table, and you can pick an effect with a number equal to or less than your level. A player who doesn't want the PC to have the bloodied condition can choose not to invoke it.
Here, however, I'd put the brakes on; for a few reasons:
1. PCs generally have enough going for them already, and players have enough (or too much) to keep track of as it is without adding more.
2. It's inevitable, due to player-driven design pressures, that any and all PC bloodied conditions will end up being beneficial to the PC, resulting in nothing more than an overall power boost. That said, if PCs took a hit somehow because of becoming bloodied, or even if there was some sort of benefit-penalty trade-off, I could maybe sort of get behind it. Otherwise you could easily end up with PCs intentionally trying to get hurt (or hurting themselves!) in order to invoke their bloodied condition, which seems a bit ridiculous.
EDIT to add: with one exception - I'd say a Barbarian cannot rage unless it has taken damage within the last minute, which preserves various Barbarian tropes.
To make becoming bloodied carry any sort of penalty would, however, require a fundamental change in what seems to have become WotC's overall design philosophy (which largely eschews penalties) - I'm not holding my breath on that.