Uber's name means "above," and the founders have said little about how they picked it.
The word is German. Etymonline traces German über and English "over" to the same Proto-Germanic root. Merriam-Webster lists the English prefix as coming from German über, with the sense of "super": a superlative example of its kind, or something to an extreme degree. Old High German had ubar, an ancestor of the modern word.
In German the word has long been ordinary. It is a plain preposition, and it also works as a prefix, as in überdurchschnittlich, "above average." In English, though, it carried baggage. The Boston Globe traces it through Nietzsche's übermensch, which English readers came to call "superman" after George Bernard Shaw's 1903 play. It notes that after the Second World War the word was tied to Nazi ideas of a master race. It was mostly used ironically, as in the Dead Kennedys' 1979 song "California Über Alles."
By the 1980s the word had lost much of that sting. It became an intensifier, like "super" or "mega," and dropped its umlaut.
The company arrived in that world. Wikipedia says it was founded in March 2009 as UberCab, with Garrett Camp developing the idea and Travis Kalanick as an adviser. Wikipedia says that in 2011, after complaints from San Francisco taxicab operators, it dropped "Cab." TechCrunch reports that San Francisco and California transport regulators sent a cease-and-desist order on 20 October 2010. It does not say the order was about the name, and other accounts date the rename to late 2010, so the timing is unsettled.
Neither Wikipedia nor the Boston Globe says why the founders chose Uber. The Boston Globe notes that taxi drivers feared the name claimed the service was "the Superman of cabs." The common explanation, that the founders wanted something superior to a regular taxi, is repeated in many places, but we found no first-hand quote from them.
So the meaning is not in doubt. The founders' intent is simply not recorded in the sources we read. The Boston Globe adds that the name has since become a verb, as in "I'll just Uber home."