As Jack attempts to resuscitate Sayid, did anyone notice anything peculiar about the way he did it?
He checks his pulse, and then there is a few seconds inbetween checking his pulse and the CPR, where he appears to sort of "hold" Sayid, in a calming, healing manner. Then as Jack proceeds with the CPR, he closes his eyes, in a deep, thoughtful, hopeful manner.
What am I suggesting? Well, I propose that Jack has some form of "healing hands" that he is currently not totally aware of. This healing doesn't work all the time, and relies on one importnat factor. "Belief", or "Hope". Jack must actually have believe and hope that he can fix this person, against all odds. If Jack has the belief and hope, then he can do it.
Look at the evidence. Against all odds, he healed Sarah, simply because he believed he could fix her. Against all odds he revived Charlie after he was hung, because he believed he could do it. Admittedly Sayids revival was slightly different, but look, Jack had that moment of deep thought and belief. If just for a moment Jack has the belief whilst making contact with his subject, then that is enough to revive them.
This all relates to Jack's conversation with Chritian about having "hope", in the episode 'Man of Science, Man of Faith.'
CHRISTIAN SHEPHARD: You might want to try handing out some hope every once in awhile. Even if there's a 99 percent probability that they're utterly, hopelessly screwed, folks are much more inclined to hear that 1 percent chance that things are going to be okay.
JACK: Her spine's crushed. I tell her that everything's going to be okay -- that's false hope, dad.
CHRISTIAN SHEPHARD: Maybe. Maybe, but it's still hope.
Could it be that these wise words from Christian were the catalyst for Jacks healing ability? If Jack follows these words of advice, and has hope and belief that he can heal, then he can heal, against all odds? Perhaps these words from Christian were the catalyst for Jacks hope, but I believe something else is going on too. Remember when Jacob visited Jack? I think that when Jacob touched our main characters at some point in their lives, he was somehow giving them all a part of himself. Jacob knew he was going to die in 2007, so he set events in motion, but he also helped leave part of his abilities behind.
Jacob gave Jack the ability to heal. However, Jacks ability to heal had to evolve over time, and had to coincide with his belief in healing. So it stands to reason that when Jack doesn't really believe he can heal, then he fails. He failed with Boone, and with Gabriellas father, perhaps because deep down he didn't truly believe those people could be fixed. Yet when Jack has hope, he can miraculously heal.