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)
You take damage at the start of your turn.
You make a saving throw at the end of your turn, unless something or someone grants you a saving throw. For example, Boots of Free Movement which have the following power:
Encounter (Minor Action)
Make a saving throw against a slow, immobilize, or restrain effect that a save can end.
Note that you cannot cheat by delaying your turn. If you are slowed until the end of your turn, it will last until you take your actions. Here is an excerpt from description of delaying which contains some other loophole closing rules as well:
Start of Your Turn: At the moment you delay, carry out the start of
your turn normally.
End of Your Turn: You don’t have a normal end of your turn. Instead,
the things you do at the end of your turn happen at two separate
times.
End Beneficial Effects when You Delay: At the moment you delay, end
effects that last until the end of your turn and that are beneficial
to you or your allies. For example, if on your previous turn you
stunned an enemy until the end of your next turn, the stunned
condition ends. You can’t prolong a beneficial effect by delaying.
End Sustained Effects when You Delay: You can’t sustain a power if you
delay. At the moment you delay your action, the “check actions spent”
part of the end of your turn occurs. Because you haven’t spent an
action to sustain any active powers, sustainable effects end.
End Harmful Effects after You Act: After you return to the initiative
order and take your actions, end effects that last until the end of
your turn and that are harmful to you. For example, if an enemy
weakened you until the end of your next turn, the weakened condition
ends. You can’t avoid a harmful effect by delaying.
Make Saving Throws after You Act: After you return to the initiative order and take your >actions, you make saving throws against effects on you.
Published in Player's Handbook, page(s) 288.
Also note the following about the "failed saving throws" (we didn't know this and it would have made a big difference last session):
Failed Saving Throw
Sometimes an effect changes as a target fails
saving throws against it. The new effect, specified in a “First Failed
Saving Throw” or a “Second Failed Saving Throw” entry, takes effect
after the target fails a saving throw against the previous effect at
the end of the target’s turn. The effect doesn’t change if the
creature fails a saving throw against it at a time other than the end
of its turn.
Published in Dragon Magazine 373.
Best Answer
Make Heal DDI checks.
Even if a character has no healing surges left, any other character can try a Heal check (DC15) to Stabilize him. If successful, no further death saves need to be made.
If they do have a healing surge left, it is a mere DC10 to allow them to use their Second Wind DDI.
Both of these checks are very makeable for even 1st level characters and should be attempted whenever there isn't immediate danger to the conscious character in my opinion.
Spread Healing Responsibilities Around
If this is occurring during combat, your group may want to ensure that most characters are able to heal others. I feel that nearly all non-striker characters should have some form of healing available. It is better to restore a dying comrade to fighting strength than to inflict damage upon the enemy.