In a recent session, one of our party members was petrified by a medusa and then kidnapped. When the rest of the party succeeded in fighting their way to rescue our party member, the statue had been disfigured. One of the character's hand had been broken off.

Assuming that the character has not been killed outright by the damage, would they be missing a hand when they are de-petrified?

Or generally, to what extent does damage to a petrified character carry over, beyond the abstraction of HP damage?

Yes, any part of the body broken off while stoned is still lost when restored

The Flesh to Stone spell states:

If the creature is physically broken while petrified, it suffers from similar deformities if it reverts to its original state.

Likewise the Basilisk entry in the MM states:

Unfortunately for [creatures turned to stone], any parts lost in stone form remain absent if the creature revives. Revivification using [oil made from a basilisk's stomach] is impossible if a vital part of the petrified creature, such as its head, is detached.

Several other creatures and effects can inflict the petrified condition (Medusas, Beholders, Gorgons, Prismatic Spray/Wall) but none of them mention any lasting effects of restoration (or the lack therof). Since there is no situation that explicitly runs contrary to the above 2, this leads me to believe that losing a hand (or paw) while turned to stone will always result in missing that hand if you are restored.