Haughty is another word for arrogant that begins with the letter H.
Hazel Hockey
"Hidalgo" is another word for a Spanish gentleman that begins with an 'h'.
This could be haggard.
Hubble telescope is a space word. It begins with the letter h.
Hexagon (:
Hindustan is another word for India (literally meaning "land of Hindus").
habitation
Homophobes?
happen
Half
haughty