Dramatic Death and Last Stand
Laying about, rolling a d20 every round and being silent is not the dramatic fantasy death most of us have in mind for our PCs. No opportunity for last words or a final act - anticlimactic more often than not. It feels bad as both a player and a DM.
This set of rules hopes to provide opportunities for heroic and narrative actions by replacing the rules for what happens when you hit 0 hit points.
As I Lay Dying
If damage reduces you to 0 hitpoints, and fails to kill you outright, you fall prone and are stunned.
You are dying, but semi-conscious and able to barely speak -- perhaps to pass on one last piece of wisdom or a secret.
Going Out on Your Terms
While at 0 hit points, before you would make a death saving throw on your turn, you may choose to make a Heroic Stand.
You cannot make a Heroic Stand if you are unconscious, paralyzed, petrified, or have otherwise lost control of your body.
If you choose not to make a Heroic Stand, you fall or remain prone, are stunned until the start of your next turn, and must make a death saving throw, as normal.
Heroic Stand
If you choose to make a Heroic Stand, you immediately fail your death saving throw, are no longer stunned, and can perform your turn as normal.
You cannot be regain hitpoints or stabilized in any way, short of a Wish spell, until your Heroic Stand ends at the start of your next turn.
Final Act
If you are currently making a Heroic Stand and reach three failed death saves you may make one action immediately. Once the action is complete, you die.
Optional Variant Rules
Here are some suggestions to help fit the tone of your table.
Challenging Heroic Stands
When you attempt to make a Heroic Stand you must succeed on a DC 10 constitution (or wisdom?) saving throw, or you fail to make a Heroic Stand, remain stunned, and must make a death saving throw as normal.
Hobbled Heroic Stand
When making a Heroic Stand your speed is halved.
Defensive Stand
When making a Heroic Stand you are unable to make attacks or cast spells.
Others Can Heal You
When making a Heroic Stand only your own sources of regaining hitpoints or stabilization are ineffective.
Design Notes
As I Lay Dying
This gives players a small opportunity to give their final words, which is great for narrative, as well as choose to make a Heroic Stand.
There is also a somewhat large and hidden change here. Normally when you hit 0 hit points you are rendered unconscious instead of stunned. These effects are quite similar, but when you're unconscious you also can't speak, are unaware of your surroundings, and (here's the big one) any attack that hits is a critical hit if the attacker is within 5 feet, which means 2 failed death saving throws for the dying. These rules mean those auto-crits would generally no longer happen.
For my table this a feature not a bug, but others might find losing the 5e version of the coup de grace swings the balance too far in the favor of the players. In my experience, enough monsters have multiattack to make it easy enough for them to finish off any player in a round or two without the coup de grace, and auto-crits made it too easy.
Going Out on Your Terms
First, it's valuable to recognize that making a Heroic Stand is entirely optional. If players choose not to make a Heroic Stand essentially nothing changes for them (except the above noted change to auto-crits).
The ability to take a turn while dying is balanced against the cost a failed death saving throw and being unable to heal or be stabilized for a turn. The second part, the inability to be healed, is absolutely essential - without this, every player character would be able to just down a healing potion or cast a spell and essentially negate most danger of dying. Yes, this is a design choice with no convincing explanation within realism; this small loss of verisimilitude is a trade-off for gains in player agency and heroic drama that I think fits well in 5e. As always, your mileage may vary - do what's right for your table.
Credits
"As I Lay Dying" was borrowed nearly word-for-word from a Twitter post from @litzabronwyn.
"Going Out on Your Own Terms" is adapted from a homebrew post by u/Solemnpancake