OK, now this has my attention. A few quick thoughts...
1) bloodied HP (BHP = [STR mod + Con Mod x size]) in addition to HP as normal
Maybe I'm just dumb, but what are you using to get a number for "size" to use as a multiplier? I ask because I'd like to get an idea of how many BHP a typical character would have.
2) Armor w/ DR (only when attacking BHP: DR = AC-10), this makes heavy armor more important which we like
3) death at 0 BHP
Death outright, or death saves as per 5e RAW?
And, do you have any mechanism for going unconscious without dying?
One that leaps to mind would be any time you take damage that leaves you in positive BHP you need to make a Con save at DC [5 + the amount of BHP you have lost from your total] or collapse unconscious.
Example: character has however many normal hit points (I'll call these Fatigue Points, or FP) plus 14 BP. Character takes damage that leave it with 12 BP, meaning it's lost 2 thus its consciousness save is DC [5 + 2] 7. Character then gets hit again for 9 more points damage leaving it at 3 BP; it now has to make a save at DC [5 + 9 + 2] 16 to stay upright.
Unconscious characters cannot benefit from second-wind-like effects, nor can they heal themselves in any way, but can be healed as normal by someone else (but see below!), or awoken without healing if someone spends an action to do so and is within touch reach.
If not awoken by combat's end the downed PC makes a save of the same DC that knocked it out, success meaning it awakens on its own and failure meaning the character will bleed out and die in [some time period based on how damaged it is] if not tended. (most of the time someone will tend the fallen PC right after combat anyway, making this largely moot; this mechanism exists for cases where there's no-one left to help e.g. everyone is dead, unconscious, or fleeing for their lives) EDIT TO ADD: For added interest, if on such a roll the result is 1 under, 1 over, or bang on the DC number then nothing happens, the character remains unconscious but stable, and the process repeats in 1d30 minutes.
5) regular healing for HP, slow healing for BHP (1 HP per week)
To make BP damage even more significant you could also have magical healing of any kind be only half as effective when curing BP.