My thought process with this rule is that I always thought it was weird that a character is instantly knocked unconscious when dropped to 0 hit points, even if they were fighting to the fullest of their ability on the brink of death but a moment before. Did that paper cut really hurt so much that you passed out?

Thus, I devised this rule, a kinda buffer between being totally combat capable and being completely worthless. You're on the ground and may be free from the ire of whatever you were attacking, but you can use the last of your strength to try and stabilize yourself or do something that may turn the tide of battle while the enemy's attention is off of you.

At 0 hit points, instead of becoming unconscious, a character is 'downed.' Their speed drops to 5 ft., they fail all Strength and Dexterity saving throws and checks, and they incur the effects of being prone. The only actions a downed player can take are Cast a Spell, Dash, Hide, Improvise, Ready, Search, and Use an Object. While downed, a character can still cast spells, but they can only cast ones with only verbal components.

A downed character is still somewhat in critical condition, and must make death saves. However, characters now make death saves at the ends of their turns instead of at the beginning, because of the ability of characters to take a limited selection of actions while downed.

A downed character who has not yet made any death saves can choose to not take any actions or bonus actions during their turn to 'gather their strength.' If they choose to do this, they have advantage on their first death save. If they succeed, they rise at the start of their next turn with 1 hit point, incurring a point of exhaustion to fuel their second wind. This is to retain the usefulness of certain features such as the Half-Orc's Relentless Endurance. A creature cannot do this if they have 2 or more points of exhaustion already.

At two failed death saves, or after taking more damage after becoming downed, a downed character goes into shock and becomes fully unconscious.

A creature made stable is still downed, and requires healing to be brought back up to a fighting-capable capacity. While stable, a creature cannot 'gather its strength' but will successfully regain its 1 hit point after 1d4 hours. An unconscious creature made stable remains unconscious.

Effects that instantly kill characters still successfully do so, in whatever gruesome way they describe.

I haven't implemented this rule yet, I just kinda stormed it up while thinking of plot hooks and decided to post it here for some outside opinion.

I think it's cool, especially for campaigns with no clerics to cast spare the dying. I would personally only let them take the hide or use an object action. Also, I might say that if they do this a number of times equal to their con bonus, then after that they have disadvantage on the first save. If they do it again then in second save etc. 

The first thing that comes to mind is that this mechanic is even though it's interesting it might also run into conflict if you have a character who's playing a zealot barbarians. Reason being, their subclass capstone allows them to remain conscious while at 0 hit points. Now with how you have this feature worded, it can still work because it would allow the zealot barbarian to do their turns normally as long as they remain raging. So, I think it should work, but be careful if you have someone playing a zealot barbarian.

Quote from FallenPhoenixIV >>

The first thing that comes to mind is that this mechanic is even though it's interesting it might also run into conflict if you have a character who's playing a zealot barbarians. Reason being, their subclass capstone allows them to remain conscious while at 0 hit points. Now with how you have this feature worded, it can still work because it would allow the zealot barbarian to do their turns normally as long as they remain raging. So, I think it should work, but be careful if you have someone playing a zealot barbarian.

 In that case, I would use specific beats general. So zealots would not drop with this. 

I like the idea behind this, but I'd say try and keep it as simple as possible; let them move five feet, Hide or Use Object (to down a healing potion, get through a door or such) and then the death saves bit (stay still for advantage, unconscious after two failures).

Basically they should be absolutely focus on saving themselves and nothing more.

That said, I like the idea of being able to make a desperate last-gasp, so maybe let them make a single attack or cast one spell, but doing so immediately fails a death save unless the attack/spell healed them, and no matter what they fall unconscious (so even if they managed to heal themselves they are unconscious but stable). This would be useful if you go down and you know your own party members aren't far behind you, or won't be able to help you, and that one last hit or spell might make all the difference.