What sort of actions are you permitted to take when you are dying? Can you activate powers that might help you make the save? Also, is getting less than a 20 on a death save considered to be "failing the save"? I ask because we had a character who was dying, and got an 18 on the roll. The player then activated her "Heroic Effort" power, which takes "No Action" and grants "+4 to a failed saving throw". Does that work? Also, is there any way you can leverage unspent action points for a bonus to your death saves?
DyingDDI is a condition, part of which states that you are unconscious.
UnconsciousDDI is a condition as well, part of which says that you can take no actions. Therefore you cannot activate powers unless they explicitly state that no action is needed.
The Death Saving throw is still a saving throw, you only fail if you roll under a 10. Rolling a 20+ is a special exception that allows you to spend a healing surge. Heroic effort can only be used on a failed attack or a failed saving throw. Therefore it could not be used in this situation. (I'd have no problem with a DM allowing it though, it sounds like a cool use to me). Note that it could be used on a roll of 9 or less (a failed saving throw)