The OED’s ancient (1888) etymology for brothel goes as follows:
Etymology: Middle English broþel, < Old English broðen ruined, degenerate, past participle of bréoðan to go to ruin: a variant of brethel n. ["A worthless fellow, good-for-nothing, wretch."]
The modern sense arises from confusion with an entirely different word bordel n. (q.v. [ < Old French bordel ‘cabin, hut, brothel’]); the brothel was originally a person, the bordel a place. But the combinations bordel-house and brothel’s house ran together in the form brothel-house, which being shortened to brothel, the personal sense of this word became obsolete, and it remains only as the substitute of the original bordel.
That’s still about as much as can confidently be said, but there are a couple of weaselly bits there (right at the start the punctuation between the two forms in “broþel, < Old English broðen,” and later on “a variant of”), and etymologist Anatoly Liberman, ever eager to explore new terrain, sinks his teeth into them in this OUPblog essay. Among the interesting things he has to say is that bordel “existed in Old French”:
Its root (bord-) is a Germanic word, akin in sound and meaning to English board. From an etymological point of view, bordel designated a small board house, a hovel (-el is a diminutive suffix) and only later acquired the meaning that has stayed without change to this day.
He goes on to say “I would risk defending and developing an etymology offered in The Century Dictionary but disregarded by all later authorities,” and although I don’t necessarily buy it, it makes enjoyable reading (though Liberman’s odd puritanism can be offputting; he talks about “the unhealthy popularity of our F-word in the remotest countries of the planet”). Anyway, read the whole thing if the topic is of interest to you; thanks for the link, Kobi!