No, Thomas Crapper did not invent the flush toilet and thus give his name to posterity. The word crap, meaning excrement, is from the Old French, via Middle English, crappe, which stood for the grain that was trodden underfoot in a barn. The word originally derives from the Latin crappa.
But there was a Thomas Crapper, and he did improve, but not invent the flush toilet. He was an Englishman, 1837-1910, who invented the ball and suction device (British Patent # 4,990) found in modern toilets that allows an efficient flush with an minimum of water and also shuts off the flow to the tank once refilled. This was a case where an appropriately named man made a contribution, not a case of an eponym. The OED2 traces its use at least to 1846, crapping ken for a water closet. Since Crapper did not invent his version of the toilet until 1882, it is obviously not the origin.
Incidentally, the word craps, for the dice game, derives from the word crab and is unrelated to the term for excrement. It is a French corruption of the English term which stood for a throw of two or three. Why the English called such a toss crabs is not known, but according to the OED2, it dates at least to 1768. The tale that it derives from the nickname of Bernard de Marigny, a New Orleans gambler (circa 1800) known as Johnny Crapaud, literally Johnny Toad, is fanciful, but not the correct etymology.