Custom name for bonfire does not persist.
I am not sure what causes this as it is mostly there when I log on.
But at random, when I go to light it, it has reverted to the default name.
This only happens with the bonfire.
Chests and furnaces which I also give custom names, have always kept their name.
My stoves (normal, and improved) do that. Names will not stick. It did the same with my large bonfire, but after a while it seems to have worked. ( I am on PS4 though) There is a user here who says the same thing happens with their flower pots.