The swinhoei epithet in the species' binomial name comes from the name of Robert Swinhoe, a British consul in China and an avid naturalist, who sent a specimen of this turtle from Shanghai to John Edward Gray, the top turtle expert at the British Museum. The latter, accordingly, named the new species Oscaria swinhoei. The Oscaria in the binomial species name was replaced with the Rafetus at some later point, when zoologists reclassified the species to the Rafetus genus. But what does the genus name, Rafetus, actually mean?
Although Latin in appearance, the etymology of the name Rafetus was somewhat baffling to the volunteer Latinists at Yahoo Answers, when somebody raised that question there a few years ago. So one has to go to the primary sources. Before the Yangtze turtle was re-classified as a Rafetus, the Rafetus genus was "monospecific", i.e. contained only one species, the Euphrates Softshell Turtle, Rafetus euphraticus. While the Euphrates Turtle had been known to Western science since 1797, it got its present Latin name, only in 1864, from the same Gray who, based on his analysis of its skeleton in the British Museum, decided to establish a separate genus for it. Somewhat surprisingly to me, Gray's paper - or his earlier monographs on the related subjects - did not explain the meaning of the Latin name for this genus, or, for that matter, for any other genus he was describing. (Were such things supposed to be transparent to the zoologists of the day? Who knows...) In any event, the name of the genus Rafetus that Gray established in 1864 was clearly based on what he reported as the species earlier names, viz. Testudo rafeht and, later, Trionyx rafeht. (Testudo was the original, Linneus', genus for all turtles; Trionyx, a later established genus for all soft-shell turtles). Gray would also use "the rafeht" as the species' supposed common name.
But... what did "rafeht" mean? For this, we have to go to what is the real primary source, i.e. the Voyage of Guillome-Antoine Olivier. Olivier was the French naturalist who avoided the unpleasant fate of Lavoisier by being sent, soon after the French Revolution, as an envoy to the Ottoman Empire and Persia. In 1797, when crossing the Euphrates on his way back to France, he shot a large turtle, which, he said, the Arabs called rafcht. (As he wrote in French, his ch must stand for a sound more like the English sh). So the species became known as Testudo rafcht, with "rafcht" actually used as a "common name" in some publications of the time. Gray apparently misread "rafcht" as "rafet"... which I would not find entirely surprising.
And what does rafcht (rafšt, in a more standard Arabic transcription) mean? Arabic dictionaries don't have such a word, but they do have rafš ( رفش ) , which means "spade" or "shovel". And, apparently, ar-rafš (الرفش), i.e. "the spade/shovel" is still used as the common name of this creature, according to e.g. to this leaflet of Wildlife Middle East (which also has an English version). So we've got here, ultimately, a (Contemporary) Latin word of Arabic origin!
Well, the shell does look like a shovel! (Photo by Reqê Firatê)
P.S. Any actual Arabic speakers are welcome to comment!