He is already old, though he hears the whole world is new, when the Granger woman shows up with the kit. House-elf slavery remains legal, and he still belongs to Harry Potter, though the great wizard resides elsewhere now, and has left Kreacher with nothing but the single, standing order that he is to keep Grimmauld place well maintained and prepared for emergencies. It occurs to Kreacher that Master Potter probably sees this as a kind of freedom. If Kreacher will not accept clothing (and he will not, thank you very much!), then he will at least be given complete rule over his domain of relics.
Elf-slavery remains legal, but the trafficking of elvish young has been banned by Ministry statute (baby-steps, sighs the Granger woman, over a cup of tea and one of his best scones), and, despite the best efforts of the S.P.E.W's volunteer corps, not all of the liberated (confiscated) young ones can be returned to their parents. And so, the Granger woman arrives at the doorstep of Number 12 Grimmauld Place with a newborn kit in her arms, all giant eyes and curious, swiveling ears.
Kreacher has never had anything to name before. His own name was bestowed on him by Master Arcturus: "Another litter of those creatures running about, eh?", "You there, creature, bring that here", "Stupid creature, do as you're told!" It was young Master Regulus, clever young Master Regulus, who'd taught him how to spell it, with a K at the front and an R at the end. Kreacher had supposed it was by tradition that his name was the same as his mother's, and her father's, and both of his uncles'. Mistress Walburga (rest her soul) was so very fond of tradition.
Kreacher has heard that each goblin has their very own, secret name, names that no wizard has ever heard or spoken. He wonders if house-elves should have such things, but elves, unlike goblins, have no language of their own. There are no words of theirs unshaped by human mouths. Besides, he's had too many secrets in his life.
He names the infant Reggie.
She grows quickly, like all elf children, and within a few months she is following him all round the house, watching goggle-eyed as he scours and dusts. He teaches her nearly everything he knows: how to clean, and cook, the sacred history of the house they keep. He teaches her everything but servitude. She has never had a master; he knows she never will. Once he did not think such a thing was right. A house-elf without a master—it was certainly against nature! But it feels right, feels natural, for her.
She calls him Grandfather; it's the best name he's ever had.