It was Lily’s enchantment, which was carried by Harry’s blood.
Lily’s enchantment being in the Dark Lord was what tethered Harry to life. Harry’s blood was only relevant because it carried this enchantment in it, so the Dark Lord using it to rebuild his body also means that along with Harry’s blood he’d have taken some of that enchantment.
“Without meaning to, as you now know, Lord Voldemort doubled the bond between you when he returned to a human form. A part of his soul was still attached to yours, and, thinking to strengthen himself, he took a part of your mother’s sacrifice into himself. If he could only have understood the precise and terrible power of that sacrifice, he would not, perhaps, have dared to touch your blood … but then, if he had been able to understand, he could not be Lord Voldemort, and might never have murdered at all.”
- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)
The Dark Lord kept Lily’s enchantment alive within himself because he’d used Harry’s blood containing it, which tied Harry to life because Lily’s enchantment and its protective power over Harry still survived in the Dark Lord’s body. J.K. Rowling explains this in the F.A.Q. of her website.
Having taken Harry’s blood into himself, Voldemort is keeping alive Lily’s protective power over Harry. So Voldemort himself acts almost like a Horcrux for Harry – except that the power of Lily’s sacrifice is a positive force that not only continues to tether Harry to life, but gives Voldemort himself one last chance (Dumbledore refers to this last hope in chapter 35). Voldemort has unwittingly put a few drops of goodness back inside himself; if he had repented, he could have been healed more deeply than anyone would have supposed. But, of course, he refused to feel remorse.
- F.A.Q. - What exactly happened when Voldemort used the Avada Kedavra curse on Harry in the forest? (J.K. Rowling’s website)
Since the Dark Lord kept a part of Lily’s enchantment alive even when Harry was hit by the Killing Curse, Harry was tied to life externally in a way similar to if he’d had a Horcrux. Dumbledore refers to this when talking to Harry in King’s Cross - it was Lily’s enchantment that kept him alive.
“He took into his body a tiny part of the enchantment your mother laid upon you when she died for you. His body keeps her sacrifice alive, and while that enchantment survives, so do you and so does Voldemort’s one last hope for himself.”
- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)
The enchantment was the reason why Harry survived - if his blood didn’t carry the enchantment, he’d have died. If the Dark Lord had used the blood of another of his enemies, one whose mother didn’t die to save them, he would still be able to kill them despite having used their blood - just taking someone’s blood wouldn’t be enough to tether them to life.