When the word omelette was first used in an early seventeenth century French-English dictionary, it was spelled omelet, and later forms included ormlet, omlet, omlette, and omellette. As you may surmise, it was borrowed from French, where it had a very interesting history. The word was metathesized (the m and l switched places) from French alemette, which was a diminutive of the word for "knife blade", alemele. Before that, it's thought that alemele was an accidental rebracketing of la lemelle (meaning "small plate") because of people mistaking it with l'amelle. Finally, lemelle derives from the Latin noun lamina, also "thin plate" (with that thought to trace to the Proot-Indo-European reconstruction stelh​, "broad"). Usage of the word omelette rapidly increased in the early 2000s and peaked in 2017.

The word syllabus was first used in English in 1656 by lexicographer Thomas Blount in a dictionary listing out a bunch of difficult loanwords. Then, it was described as "a Table or Index in a Book, to shew places or matter by Letters or Figures". Before that, syllabus was used in Medieval Latin, and that was taken from Ancient Greek sittybos. The reason for the sound change is quite interesting: it was supposed to be syttabus, but some fifteenth-century scholar translating Cicero's Ad Atticum misprinted the ts as ls, probably because they confused the word with the unrelated Ancient Greek verb syllambano, "to put together". The real etymon is unknown, but it probably had something to do with parchment. Unsurprisingly, Google Trends shows searches for syllabus peaking in August and January - the beginning of school semesters.

Mountweazel is a rather delightful word describing the practice of inserting fictitious entries into works of reference to catch anybody plagiarizing them. Examples include esquivalience ("an avoidance of responsibilities") in the New Pauly encyclopedia, prothesize ("place before") in several Merriam Webster dictionaries, jungftak (a type of Persian bird) in the New Oxford American Dictionary, and various fake  "paper towns" on maps. All these entries exist to serve as copyright traps that prove that someone is stealing the work. The term mountweazel is named after Lillian Virginia Mountweazel, a fake person listed on page 1,850 of the 1975 edition of the New Oxford American Dictionary as being an American photographer who died in an explosion.  Since its first attestation, usage of mountweazel rapidly increased around the turn of the twenty-first century, peaking in 2010.

The word flannel was first used in a 1503 ledger detailing the expenses of Elizabeth II, the wife of King Henry VII. There, if was spelled fflanell, and other spellings at the time included flannen, flannion, flannin, flanning, and flanell. Because of the structure of the word and the etymological timeline, the Oxford English Dictionary suggests that fflanell comes from Welsh gwlanen, which also meant "flannel". That would come from their word for "wool", gwlan, which ultimately derives from Proto-Celtic wlana and Proto-Indo-European hwlhneh, both also meaning "wool". An alternative origin proposed by some linguists who disagree with the theorized sound changes is that flannel comes from the Old French noun flaine, describing a type of wool. For now, I'll stick with the OED explanation.

The earliest attestation of the word tequila that we can find is from an 1844 diary entry by American merchant Josiah Gregg while he was exploring, to describe the drink in context of the eponymous region of Jalisco, Mexico, which was renowned for producing lots of high-quality tequila. That was borrowed from Spanish (where it was occasionally also spelled tequilla and tequela), and the Spaniards got the word from the Nahuatl noun tequillan​.  There are several theories as to where this derives from - proposed translations include "place of work","place of tricks", "rock that cuts", and a diminutive of "breast" - but the most widely accepted meaning is "place of tribute". The term tequila sunrise to describe the cocktail with orange juice and grenadine first started cropping up in 1965, and tequila sour is from 1966.