The name of the flu is a horoscope. Influenza is simply the Italian word for "influence," and the influence in question came from the stars.
The idea goes back to Latin. Influere meant "to flow into," built from in- and fluere, "to flow." Medieval astrologers used influentia for an invisible power that streamed down from the planets and stars. Depending on their positions, it was thought to shape a person's character, fortune and health. English picked up "influence" in this astrological sense in the late 1300s.
Italians took the next step. If the heavens could shape a life, they could also bring sickness. According to Etymonline, Italian was applying influenza to diseases by 1504. One example is influenza di febbre scarlattina, for scarlet fever. The word covered illness blamed on astral, occult or atmospheric influence. It was an explanation for outbreaks that seemed to arrive from nowhere and hit everyone at once.
English borrowed the word in 1743. That year an epidemic began in Rome and spread across Europe. The Gentleman's Magazine reported in April that "a Sort of Plague has broke out" in Rome, adding "they call it the Influenza." English readers took the Italian label as the disease's name. The theory of starry influence faded, but the name did not.
For a long time the word was used loosely. Etymonline notes that by the mid-1800s people often applied it to any bad cold. Only later did it come to mean one specific viral infection.
The short form has its own oddity. English usually clips words at the front or the back: phone from telephone, bus from omnibus. "Flu" instead keeps the middle syllable and drops both ends. Etymonline records it as "flue" in 1839 and in its modern spelling by 1893, and calls this kind of shortening uncommon.
So a doctor diagnosing the flu is, word by word, blaming the stars. The same Latin fluere runs through fluid, fluent and influx, all about things that flow.