The word atom is a promise that the thing cannot be cut. Physicists broke that promise in the early twentieth century, and the name stayed anyway.
It comes from the Greek atomos, "uncut" or "indivisible." The word joins a-, "not," to tomos, "a cutting," from the verb temnein, "to cut." In the fifth century BC the philosophers Leucippus and Democritus argued that matter was made of tiny solid particles moving through empty space. Those particles could not be divided or changed, so "uncuttable" was the natural name.
The idea was not popular with everyone. Aristotle rejected the notion that nature could be reduced to atoms drifting in a void. The Romans kept the word alive as atomus, used especially by the poet Lucretius. English picked it up in the late 1400s for a hypothetical, very small body that could not be divided.
For a while the word also measured time. In late classical and medieval use, an atom could mean the smallest unit of time, with 22,560 of them in an hour, according to Etymonline.
The scientific career began with John Dalton, who revived the word for chemistry. Etymonline dates his first use to 1805. His A New System of Chemical Philosophy of 1808 speaks of "the ultimate particles or atoms" of bodies. He preferred "atom" to other terms because, in his words, it "includes in itself the notion of indivisible."
That notion did not last a century. J. J. Thomson's cathode-ray experiments in 1897 showed that atoms contain much smaller charged particles. In 1899 he concluded that producing negative electricity "involves the splitting up of an atom." Ernest Rutherford then found a dense nucleus at the center, and in 1917 he knocked pieces off nitrogen nuclei with alpha particles.
So the atom turned out to be very cuttable. The name survives as a fossil of an older guess. The same Greek verb for cutting still works honestly in other English words, from the dissections of anatomy to the cut-down summary of an epitome.