haemorrhoid: Greek haima = blood, and rhoia = to flow, hence likely to bleed.
hallux: Latin hallex = great toe (hallucis = of the great toe).
hamate: adjective, Latin hamus = a hook, hence, hooked.
hamstrings: the tendons of the muscles of the ham - i.e. of the back of the thigh - felt behind the knee when the leg is flexed against resistance (semimembranosus, semitendinosus and biceps femoris).
helicotrema: Greek helix = a coil, and trema = hole, hence the aperture at the apex of the bony cochlea whereby scala vestibuli communicates with scala tympani.
hyperacusis: Greek hyper = over, and akousis = hearing, hence excessive sensitivity to sound.
hypoglossal: adjective, Greek hypo = under, and glossa = tongue.
hypophysis: Greek hypo = down, physis = growth, hence, a downgrowth (from the brain). However, this is not the whole truth. Part is an upgrowth from the pharynx, adjective - hypophysial.
hypothalamus: Greek hypo = under, and thalamus (q.v.), refers to part of diencephalon.