hypochondria (n.)
"unfounded belief that one is sick," by 1816; a narrowing from the earlier sense "depression or melancholy without real cause" (1660s); from Middle English medical term ipocondrie "lateral regions of the upper abdomen" (late 14c.). This is from Late Latin hypochondria, from Greek hypokhondria (neuter plural of hypokhondrios), from hypo- "under" (see hypo-) + khondros "cartilage" (in this case, of the false ribs); see chondro-.

The sense "morbid melancholy" reflects the ancient belief that the viscera of the hypochondria (liver, gall bladder, spleen) were the seat of melancholy and the source of the vapors that caused such feelings. The attempt to put it on a scientific bases passes through hypochondriasis. Also see hype (n.). The poet Cowper is an oft-cited example in late 18c. literature. The focus of sense on the particular symptom "unfounded belief that one is sick" seems to begin 1790s with William Cullen, M.D., professor of physic in the University of Edinburgh, who made a specialty of the topic:
A languor, listlessness, or want of resolution and activity, with respect to all undertakings; a disposition to seriousness, sadness, and timidity; as to all future events, an apprehension of the worst or most unhappy state of them; and, therefore, often upon slight grounds an apprehension of great evil. Such persons are particularly attentive to the state of their own health, to every the smallest change of feeling in their bodies; and from any unusual sensation, perhaps of the slightest kind, they apprehend great danger, and even death itself. In respect to these feelings and fears, there is commonly the most obstinate belief and persuasion. [Cullen, "First Lines of the Practice of Physic," Edinburgh, 1791]
Though to Cullen the clinical definition of hypochondria also included physical symptoms and pains as well as these mental delusions. As the old medical beliefs faded, the word dropped from clinical use but remained in popular use for "groundless morbid fear for one's health." In the 1830s hypochondria could mean merely "morbid melancholy," also "apprehension of evil respecting health, without sufficient cause," and "upper abdomen."