Technically Welsh, but
Henwen was an oracular pig (probably best known from Lloyd Alexander's Prydain Chronicles books) from some old legends who got incorporated into the Arthurian tales.
There's a tiny handful of other H-named characters from King Arthur which are
listed on Wikipedia, including a dog named Husdent, who was owned by Tristan (of "and Isolde" fame), which seems like it would be very fitting for a theme-named puppy.