Medieval Latin influentia: the ethereal fluid streaming from stars into human destiny, standard astrological physics for centuries. Influenza (the disease) was named for the same celestial inflow in 18th-century Italy. The fluere family: fluid, fluent, affluent, confluence, superfluous (overflowing), effluence.
The modern irony: influence spent centuries as an invisible fluid from the sky, became metaphor, and has now been re-literalized as an invisible fluid from the cloud, measured in reach. The influencer is the astrologer's star returned with a ring light.
Fluere's empire suggests the mind's default physics is liquid: money flows (currency, another flow word, from currere's water imagery), ideas stream, attention pools. Influence is simply the most honest case: a word that never stopped meaning fluid, only changed what flows.
If influence is a fluid, who controls the tap?