Yee-roh is the Greek answer — but not the only American one
In Greek, the word is pronounced roughly YEE-roh, with a soft initial sound close to an English "y." That is the pronunciation you will hear in most Greek-owned restaurants, and the one food writers tend to recommend. But across the United States, plenty of people say JY-roh, like the first half of gyroscope, and a smaller group says HEE-roh or something in between. All of them get handed the same sandwich.
So this is not a contest between right and wrong as much as between the original word and the word as it landed in America.
A word that means "turn"
The name is the Greek word gyros, "circle" or "turn," chosen because the meat is stacked on a vertical spit and roasted as it rotates. The dish is a close cousin of the Turkish döner kebab and the Arab shawarma — the same idea, carried around the eastern Mediterranean, each time renamed for its own language. Even the name is a translation: döner also means "turning."
The Greek letter gamma, at the start of the word, is not pronounced like an English hard g. Before certain vowels it softens into a sound close to "y," which is where YEE-roh comes from. English readers, seeing the letter g, do what English spelling has taught them to do.
Why so many Americans say "jy-roh"
The English word gyroscope, along with gyrate and gyration, comes from the same Greek root, borrowed much earlier and pronounced with a soft "j" sound. When the sandwich became popular in the United States in the 1970s, American diners already had a pronunciation for those exact letters, and they applied it.
It helped that the dish spread fast and far from its first communities. Mass-produced meat cones made it possible to sell gyros in diners, food courts and stadiums with no Greek cook in sight, and the pronunciation traveled with the menu board rather than with the people who brought the recipe.
The singular, the plural and the s
There is a second, quieter argument folded into the first. In Greek, gyros is the normal singular form; the final s is not a plural. English speakers heard the s, assumed it was a plural, and created a new singular: one gyro, two gyros. Most American menus now use gyro, which is a perfectly normal thing for a borrowed word to do.
Say it out loud before you vote. Then ask yourself whether you would say it the same way at the counter of a Greek family restaurant — many people admit they would not.
Sources: Greek etymology of gyros ("turn," "circle") per standard etymological dictionaries; pronunciations listed in Merriam-Webster; culinary histories of the gyro, döner kebab and shawarma.