This unnamed man was the father of Kate Peltzer, the husband of her mother, and the father-in-law of Billy Peltzer. He did not appear in either Gremlins 1 or 2, but was mentioned in the first one by Kate, who was evidently still distressed about his death.
He had been a very kind man and had cared about his family, going as far as attempting to go down the chimney on Christmas Eve one year and surprise them with presents. However, he had died in the process, having broken his neck and had instantly died. His death was one of the hardest experiences Kate had ever dealt with, and had made her (and her mother) decide they would not celebrate Christmas anymore. Although his last name was Beringer, it is unknown what his first name was.