Each cell is a holographic quantum entangled computer of the whole person.
The super hero in question is actually a collection of organic cellular quantum computation masses.
A typical person has about 37 trillion cells, and 7E27 atoms, making each cell capable of having an entangled particle with every other cell in the body simultaneously. (source: google)
By allowing each cell to communicate and compute the state of the whole person, any damage to the person can be undone by allowing the portion of each cell that is a quantum computer to collapse the wavefunction so that it's entangled pairs are moved to their original state before the damage occured. The network of linked particles ensures the whole person can be restored from ambient energy sources allowing the motion of the particles into their undamaged positions.
Since each cell can signal every other cell, the cells can assume a policy of restoration to the largest available undamaged portion of the whole. If two are exactly equal, a random tiebreaker can occur to decide which section to restore from. If any amount if the entangled network is completely unavailable, new tissue is grown and new particles entangled.
Any additional mass can be acquired from the surrounding environment, or reused from damaged tissue that is available.
This would impose several limits on regeneration though:
In a near energy-less or mass-less environment regeneration cannot occur, so deep space would probably take a long time to regenerate from the stray photons and interstellar particles.
If every single cell is destroyed, and thus all quantum computers capable of collapsing the wavefunction of surrounding matter are destroyed, the person dies.
If the person is violently blown apart at over 50% light speed, the particles travelling in opposite directions redshift out of each other's universes and can no longer communicate, this could lead to duplicates in this scenario, but I think is hard enough to accomplish that we can consider it impossible in normal circumstances.
The amount of mass needed in each cell to link to every other cell is noticeable. chopped off limbs would lose close to 1% of it's mass as entangled particles are drawn away and back to the regenerating body.