Per the spell description, Bigby’s Hand is an object:
The hand is an object that has AC 20 and hit points equal to your hit point maximum.
As an object, my instinct is to say that like other objects, Bigby’s Hand could only be healed by a spell like Mending which explicitly heals objects:
This spell repairs a single break or tear in an object you touch.
However, Mending has 2 obstacles. First is it has a casting time of 1 minute, the same as the duration of the Bigby’s Hand spell. So functionally I don’t know that the timing could work. Second, Mending continues to say:
This spell can physically repair a magic item or construct, but the spell can’t restore magic to such an object.
From RAI this wording might also prevent healing Bigby’s Hand since it is a conjured magical object, the healing of which might be considered “restoring magic to the item”.
I’m also not aware of any other spells used to heal objects other than Mending.
Moreover, the “conventional” healing spells seem to only apply to creatures, see e.g. Cure Wounds, Heal, Healing Word, etc.
So the question remains: is there a way to heal Bigby’s Hand?
There is no way to ‘heal’ Bigby’s Hand
All of the healing or temp HP-granting spells only target creatures, and Bigby’s Hand is not a creature.
Anything else would be homebrew, and I’d probably recommend against doing so. My experience with Bigby’s Hand is that it is a very powerful spell. Healing it would increase its power level, and I wouldn’t recommend adding more to this already powerful spell.
The Artificer’s Steel Defender does provide a specific mechanic for repair
As user Weiramon pointed out in a comment on this answer, the Battle Smith artificer’s Steel Defender does have a specific rule that applies to it that isn’t available to other classes:
Repair (3/Day). The magical mechanisms inside the defender restore 2d8 + PB hit points to itself or to one construct or object within 5 feet of it.
A Steel Defender is able to ‘repair’ an object, which Bigby’s Hand is, so this is a possibility if your table allows artificers. I would be wary to expand this outside of this class, and I’ve got general concerns with the artificer in general, but there is a possible method for repairing the hand.
It’s unclear to me if WotC meant for the defender to repair magical spell effects, but by the book, it works.