I think the Healing Surge system introduced in 4e is an improvement over how things worked in 3e. It liberated the cleric (and similar classes) from having to fully devote themselves to healing the party and gave a hard limit of how much damage a character can soak up during the course of an adventuring day.
It has it problems though that everyone else has already described quite well.
Here are what I would change:
1) Health system completely revamped. A character's health is divided evenly between Temporary Hit Points and Hit Points. Temporary hit points go back to full after a short rest and each character has the ability to use second wind in combat as an encounter ability to restore their temporary hit points to their maximum value. As soon as player takes any hit point damage, they become bloodied and their maximum temporary hit points gets cut in half. It remains halved until the character is no longer bloodied.
2) Each character has fewer healing surges; a character would have between 1 (wizard) to 3 (fighter). A healing surge represents a character's ability to recover from a serious injury. When a cleric uses a ability that restore hit points or you drink a potion, you recover the appropriate number of hit points and you lose a healing surge. No healing surges means you can no longer gain back health points for the day. You are missing half of your temporary hit points and whatever chunk of health points is gone until you can find safety.
3) Something like Cure Light Wounds can either restore someone's Y amount of temporary hit points or heal their actual hit points for an Y amount.
This creates a lot more resource management for the players. A character can use second wind to recover all of temporary hit points, but they can only do that once per fight. One good hit makes you bloodied which means all future fights you have 1/4 less total health. Given the extremely limited amount of healing surges each character has, you may decide to tough it out and stay bloodied for a while to maximize the use of your own healing surge. A wizard would be a lot more tempted to use their second wind in a fight because they have only once to get their actual hit points back in a day. A fighter could be more risk taking given they have more healing surges to spend; if a party usually faces 4 encounters in a adventuring day, they can be in decent shape for 3 of them.
Finally, this gives a distinct and clear separation between morale/near-misses/fate injuries and actual ones.
It has it problems though that everyone else has already described quite well.
Here are what I would change:
1) Health system completely revamped. A character's health is divided evenly between Temporary Hit Points and Hit Points. Temporary hit points go back to full after a short rest and each character has the ability to use second wind in combat as an encounter ability to restore their temporary hit points to their maximum value. As soon as player takes any hit point damage, they become bloodied and their maximum temporary hit points gets cut in half. It remains halved until the character is no longer bloodied.
2) Each character has fewer healing surges; a character would have between 1 (wizard) to 3 (fighter). A healing surge represents a character's ability to recover from a serious injury. When a cleric uses a ability that restore hit points or you drink a potion, you recover the appropriate number of hit points and you lose a healing surge. No healing surges means you can no longer gain back health points for the day. You are missing half of your temporary hit points and whatever chunk of health points is gone until you can find safety.
3) Something like Cure Light Wounds can either restore someone's Y amount of temporary hit points or heal their actual hit points for an Y amount.
This creates a lot more resource management for the players. A character can use second wind to recover all of temporary hit points, but they can only do that once per fight. One good hit makes you bloodied which means all future fights you have 1/4 less total health. Given the extremely limited amount of healing surges each character has, you may decide to tough it out and stay bloodied for a while to maximize the use of your own healing surge. A wizard would be a lot more tempted to use their second wind in a fight because they have only once to get their actual hit points back in a day. A fighter could be more risk taking given they have more healing surges to spend; if a party usually faces 4 encounters in a adventuring day, they can be in decent shape for 3 of them.
Finally, this gives a distinct and clear separation between morale/near-misses/fate injuries and actual ones.