"Omelet" has traveled by a circuitous route to arrive at its present form, say editors at Merriam Webster.
"Omelet" is derived from the French "omelette." The first known use of this word in English was in 1611. The French spelling is sometimes still used in English.The modern French "omelette" is an alteration of the Middle French "amelette," probably influenced by "oeuf," the French word for "egg." "Amelette" was an alteration of "alumette," which in turn was an alteration of "alumelle," meaning "blade of a sword or knife." It was in Middle French that the word acquired the sense "eggs beaten and cooked without stirring until set," since such a dish resembles a thin plate or blade.
Eventually the word traces back to the Latin "lamella," which was a diminutive form of "lamina," meaning "thin plate," a word that was also borrowed directly into English in the 17th century and now has a number of (primarily scientific) uses in which it is more or less equivalent to "layer."