Rolex does not stand for anything. The man who chose the name said it came to him from a genie, on a London bus.
Hans Wilsdorf and Alfred Davis founded Wilsdorf and Davis in London in 1905. The firm bought Swiss watch movements, put them in cases, and sold them to jewelers. The jewelers then printed their own names on the dials. Wilsdorf wanted a brand of his own.
In 1908 he registered "Rolex" as a trademark and opened an office in La Chaux-de-Fonds, Switzerland. Wikipedia sums up his requirements. The name had to be easy to pronounce in any language and short enough to fit on the face of a watch. It has also been said to sound like a watch being wound.
The famous part comes from Wilsdorf himself, fifty years later. According to Bob's Watches, he marked the name's anniversary on July 2, 1958, with his own account. He said he had tried combining the letters of the alphabet in every possible way. Then, riding a horse-drawn omnibus along Cheapside in the City of London, "a genie whispered 'Rolex' in my ear."
Nobody can check a genie, and the story is best read as a founder's tale told at a celebration. What it does confirm is that the word was invented, not built from existing words. That has not stopped people from supplying meanings.
The most common guess is that Rolex is a contraction of the French horlogerie exquise, "exquisite watchmaking." There is no evidence Wilsdorf intended this, and his own account does not mention it. Another guess is the sound of a watch being wound, which Bob's Watches calls unconfirmed. Neither is an origin in the usual sense. The only documented facts are the 1908 registration and Wilsdorf's reasons.
The name had to wait for its moment. For years many watches still carried jewelers' names. After the First World War, heavy British taxes on luxury imports and duties on the silver and gold for cases pushed Wilsdorf to move the business to Geneva in 1919. According to Bob's Watches, Wilsdorf decided in 1926, when the waterproof Oyster debuted, that every watch would carry the Rolex name.
So the brand that now signals Swiss precision was named in London, by a German-born businessman, after a word with no meaning at all.