Hagrid, Rubeus: The Keeper of the Keys and Grounds at Hogwarts, a giant of a man with a heart to match, especially when it comes to exotic and dangerous animals (cf. Norbert and Fluffy). Wand (supposedly destroyed, but strongly hinted to be hidden in his umbrella): oak, bendy, 16".
Etym: According to Rowling, "If you were hagrid-- it's a dialect word-- you'd had a bad night. Hagrid is a big drinker-- he's had a lot of bad nights."
Hair-Raising Potion: Something that includes rat tails as an ingredient.
Hand of Glory: An item associated with dark magic since medieval times. Usually said to be the hand of a thief or murderer hanged at midnight, modified to allow candles to be socketed in its fingertips. In this series, it provides light only to the holder.
Hannah Abbott: Etym: From Hebrew for "He (God) has favored me". In the Bible, the mother of Samuel, and in the Talmud, a prophetess; her prayer exemplifies successful petitions to God.
Harry Potter: Etym: Stated by Rowling on numerous occasions to just be her favorite male name.
Head Boy: A seventh-year student chosen for leadership and scholastic abilities who shares the prefects' duties.
Head Girl: A seventh-year student chosen for leadership and scholastic abilities who shares the prefects' duties.
Head Hockey: A pastime of the Headless Hunt.
Headless Hunt: A mass of headless ghosts who won't let Nearly Headless Nick join them.
Head Polo: Another pastime of the Headless Hunt.
Hebridean Black: A type of dragon native to the British Isles.
Hedwig: Harry's owl, named after someone he read about in A History of Magic.
Etym: Stated by Rowling to be a medieval saint, which would be St. Hedwig of Andechs. Another possibility is the devout but not canonized medieval woman who was crowned "king" of Poland at the age of 9, strongly supported religion and scholarship, and caused the restoration of the university of Kraków, which became the Jagiellonian University. (Here's the English root page.)
Helga Hufflepuff: Etym: From Norse, meaning "holy". Also an alternate name for St. Olga (890-969), the first recorded female ruler in Russa, the first Russian royal to adopt Christianity, and the first Russian Orthodox saint.
Hengist of Woodcroft (PS ch. 6): Was featured on a Famous Witches and Wizards trading card. Might be an actual historical person, but I haven't found any info on him.
Etym: Feminine of Henry from Old German haimi "house, home" + ric "ruler".
Herbology: A required course at Hogwarts, taught by Professor Sprout.
Hermes (CoS ch. 3): Percy's owl,, bought for him when he became a prefect.
Etym: The Greek name for the messenger of the gods, also the god of thieves, children, and travellers.
Hermione Granger: Etym: A derivative of Hermes. Used by Shakespeare in A Winter's Tale as the name of a queen falsely accused of adultery, who dies of the shock when the evidence in her favor is disregarded.
Higgs, Terence (PS ch. 12): The Seeker for the Slytherin Quidditch team in Harry's first year, replaced by Draco Malfoy later on.
Etym: From the medieval given name Hicke, a form of Richard, whose etymology is uncertain; may be riic "ruler" + heard "hard".
History of Magic: A required course at Hogwarts, taught by Professor Binns.
Hogwarts: The school for young wizards in the British Isles, housed in a massive castle with 142 shifting staircases, living paintings, numerous ghosts, and pretty much everything else that would have made your school the least bit interesting.
Hogwarts Express: The train that runs from King's Cross to Hogwarts at the beginning of the school year, and back at the end.
Hogwarts, a History: One of the books Hermione read to get ready for the start of school.
Holidays with Hags: One of Gilderoy Lockhart's books.
holly: Shrubs and trees of the genus Ilex, reputed to have protective powers. Harry's wand probably was made with English holly (I. aquifolium).
Etym: Rowling has given her reason for using it as its association with life and also with the word holy. Unfortunately, the second part is folk etymology; the word holly goes back to an Indo-European stem meaning "to prick". (Hollyhock, on the other hand, evolved from holy hock.)
Homorphus Charm: The spell with which Lockhart claims to have cured the Wagga Wagga Werewolf.
Hooch, Madam: The flying instructor at Hogwarts, and usually the referee for inter-house Quidditch games.
Etym: No etymology. There was a minor Dutch painter named Pieter de Hooch, or Hoogh, or Hooghe (1629-1684).
Hopkirk, Mafalda (CoS ch. 2): Someone in the Improper Use of Magic Office.
Etym: From Hopekirk, named from Northern Middle English hop(e) "valley among hills" + kirk "church". A former Hufflepuff?
Hornby, Olive (CoS ch. 16): A girl who particularly tormented Moaning Myrtle and was haunted by Myrtle for it until the day she (Olive) died what seems like an awfully early death.
Etym: Name of several places in northern England, from the Old Norse name Horni "horn" + býr "farm, settlement".
Horseback Head-Juggling: Another pastime of the Headless Hunt.
house-elf: A magical creature, bound to a specific wizard family or place, which does housekeeping tasks. If a house-elf's employer hands a piece of clothing to them, the elf is considered dismissed.
House-elves are based on brownies, mostly benevolent spirits said to inhabit particular houses or farmsteads and do chores when no one is looking. One can leave bread or a bowl of milk or cream out for the brownie to show thanks, but making any more extravagant gifts will offend it, which will turn to mischief. If a suit of clothes is made for a brownie, it will put them on and disappear.
Hover Charm: A spell used to make something float in the air. Apparently detectable from a distance (unless someone from the Improper Use of Magic Office just happened to be nearby).
Howler: An angry voice recording which plays at full volume when opened, sent by parents to give public rebukes to their children at school; a nastygram.
Hufflepuff, Helga: One of the four founders of Hogwarts.
Etym: See below.
Hufflepuff House: One of the four houses of Hogwarts, it emphasizes hard work, goodness, and self-sacrifice. The house of the saints. Badge: Or, a brock rampant reguardant, at best guess, anyway. House colors are supposed to be black and gold. The house ghost is the Fat Friar.
Etym: Invented; along the lines of "huff and puff".
Humberto: see The Great Humberto.