I think just means to clean the stove into its original shine by removing all the old burnt marks and residue. Its probably one of those phrases that came from the act of making things clean again (like "to holystone the deck of a wooden ship"). How do you manage to find such an obscure term?

I had to look this up. "Stove black" is a polish or paint that you put on wood burning stoves. It can be heated really hot and will look nice and help keep the stove from rusting. You can still buy it (https://www.lehmans.com/product/stove-black-and-polish/hearth-accessories) if you have a wood burning stove that you want to clean correctly. "Blacking the stove" would be putting that paint or polish on the stove.

Angela has the correct answer. I know this from experience.