Official confirmation from WGA leaked script confirms "Grandfather":
Forgive me. I feel it again. The pull to the light. Supreme Leader senses it.
Show me again, the power of the darkness, and I will let nothing stand in our way.
Show me, Grandfather, and I will finish what you started.
Alan Dean Foster novelization also confirms that this was basically simply a fluke of Italian translation - there's no childish "grandpa", there's only "Grandfather":
The alcove where Kylo Ren was kneeling and speaking was darker than the rest of the adjoining chambers. He kept it deliberately so, as seemed appropriate for its function. He spoke now in a tone different from the one he usually employed when conversing with others. There were no orders to be issued here, no pathetic underlings to command. The one with whom he was presently communing would understand everything Ren chose to say, in whatever voice he chose to employ. No need here and now for intimidation, for fear. Kylo Ren spoke, and the object of his words listened in silence.
“Forgive me. I feel it again. The pull to the light. The Supreme Leader senses it. Show me again the power of the darkness, and I will let nothing stand in our way.”
Alone in the room, Kylo Ren—saturnine of aspect, lithe of build, tortured of mien, and troubled of eye—gazed at the silent recipient of his confession.
“Show me, Grandfather, and I will finish what you started.”
Trembling slightly, he rose from where he had been kneeling and strode off to another portion of his private quarters. There was no response from the one to whom he had been talking: neither argument nor agreement. Only silence from the shape that had been the object of Ren’s fervor: a ghostly, deformed mask that had once belonged to another. To a figure of rumor and legend and fear.
Misshapen and malformed as it was, no one who had once laid eyes upon the countenance that had belonged to Darth Vader would ever forget it.
However, it does have some contextual aspect of a child or at least a junior apprentice addressing a senior figure, so perhaps the Italian translation wasn't as far off the mark as it seemed.