The word for the thing that keeps you dry was built to keep you cool.
Umbrella comes from Italian ombrello, which goes back to the Late Latin umbrella. That form was altered under the influence of Latin umbra, "shade, shadow," from Latin umbella, "sunshade, parasol." The ending is a diminutive, so the word is, at root, a "little shade."
Etymonline records English use around 1600, in a letter by John Donne. Merriam-Webster places the borrowing in the early 17th century. Either way, the English word arrived centuries after the object was familiar to the people who invented it.
Shade came first. According to the history Merriam-Webster draws from Britannica, umbrellas existed in ancient Egypt, Mesopotamia, China and India, where they shielded important people from the sun and signaled prestige and power. Europeans learned the idea from the Greeks, and the Romans added rain protection.
The object then faded from Europe for a long stretch. It reappeared in Italy in the 16th century, but at first only popes and clergy used it. Use spread to France in the 17th century, and by the 18th century umbrellas were common across Europe.
England is where the rain job took over. Etymonline notes that in the Mediterranean the umbrella was a sunshade, while in England it was a shelter from rain. Women reportedly carried them from about 1700, and Etymonline dates men carrying rain umbrellas to about 1750. That date is tied to Jonas Hanway, whom the 1911 Britannica says is "said" to be the first Londoner to carry one. Etymonline's text spells him "Hathaway," which is a slip. Merriam-Webster puts men's general adoption around the mid-19th century, so the timeline is loose.
The word has produced some odd relatives. Britain's clipped slang brolly is recorded by 1866. A figurative use came later: "under the umbrella of," meaning under the authority of something, is recorded by 1948.
The shadow in the root survives elsewhere too. Umbra gave English umbrage, which began as "shade" and drifted to "suspicion of being slighted," as in "take umbrage." It also gave adumbrate, to foreshadow.