It is widely accepted that the medical terminology has its roots in ancient Greek and Latin. Greek words have been used not only in the field of medicine but also in every day language for many centuries.
The actual term asthma is a Greek word that is derived from the verb aazein, meaning to exhale with open mouth, to pant.
The expression asthma appeared for the first time in the Iliad by Homer, with the meaning of a short-drawn breath, but the earliest text where the word is found as a medical term is the Corpus Hippocraticum. However it is difficult to determine whether in referring to "asthma," Hippocrates and his school (460-360 B.C.) meant an autonomous clinical entity or simply a symptom.
The best clinical description of asthma in later antiquity is offered by the master clinician, Aretaeus of Cappadocia (1st century A.D.). The numerous mentions of "asthma" in the extensive writings of Galen (130-200 A.D.) appear to be in general agreement with the Hippocratic texts.
Common terms derived from Greek include diagnosis, from the verb diagignosko, meaning to "discriminate"; symptom, from the verb sympipto, that is, "to coincide"; and the adjectives clinical and clinic, from the noun klini, meaning "bed."
I was not able to find where exactly was asthma mentioned in the Iliad but a copy of the book is embedded below, courtesy of Google Books.
Bronchial asthma in the medical literature of Greek antiquity. Marketos SG, Ballas CN. J Asthma. 1982;19(4):263-9.
Ancient Greek terminology in pediatric surgery: about the word meaning. Soutis M. J Pediatr Surg. 2006 Jul;41(7):1302-8.
Seneca called asthma “Rehearsal for Death” http://goo.gl/XrCRV
Image source: Statue of Homer outside the Bavarian State Library in Munich. Wikipedia, GNU Free Documentation License.
The Iliad, Volume 1 By Homer, Walter Leaf: