Hannah - This is a Hebrew name meaning "God has favored me", a variant of Anna. (Hannah Abbot). It has experienced much recent popularity, ranking as high as #2 from 2000-2002, and is currently (We have 2012 data people!) 22nd.
Hermione - The "interesting" name of the series, due to its use on such a main character and its former status of completely unknown. It is pronounced her-my-oh-nee. It is of Greek origin and means "messenger", related to the other name on this list Hermes. (Hermione Granger)
Helga - An old German name meaning "holy" or "sacred". This has never been in the top 1000. With other old names rising (think Ilsa and Adelaide for example), maybe it is due (Helga Hufflepuff).
Hestia - The Greek goddess of the hearth, this name has a lovely meaning and a nice sound. (Hestia Jones)
Hedwig - Another Old German name meaning "strife". Perhaps less than ideal if meaning is important to you.
Hassan - Pronounced hah-sahn, this is of Arabic origin and means handsome. It lingers in the second half of the top thousand due to its ethnic use. (Hassan Mostafa)
Hippocrates - Most people have heard of the Hippocratic oath, but I doubt many people have considered this as a given name. The meaning is technically "horse", but the main association would clearly be medicine, ethics and healing (Hippocrates Smethwick)
Hermes - The Greek messenger God, famous for his awesome sandals.
Hagrid - Okay, so this is a character's last name, but the books give it life as a first name since the character goes exclusively by this. The name references a giant in Greek mythology. (Rubeus Hargrid)
Harold - I have to admit, this name reminds me of a horror story from when I was little, but that shouldn't stop others from liking it. It is a Scandinavian name meaning army-ruler. (Harold Dingle)
Harry - The flagship name of the series. Despite the strong association with the book, it hasn't been "ruined" because there are lots of famous Harry's, most notably Prince Harry. Harry is old German for "home ruler". The bigger issue with Harry in America is pronunciation. To most American dialects, it is pronounced just like hairy. We often fail to get the strong "A" sound in names like this.
Hengist - An Anglo-Saxon name used in the Arthurian legends as a minor character. It likely is related to an old word for Stallion (Hengist of Woodcroft)