The name I keep running across is Chester A. Arthur, the 21st President of the U.S., who is widely cited as dying from complication of Bright's Disease, and this is mentioned as his cause of death in the original New York Times article announcing his death. Arthur served as president from 1881-1885 after assuming the post following the assassination of James A. Garfield. He attempted to run for re-election during the election 1884 but did not even succeed in obtaining the Republican party nomination. Apparently Arthur already knew of his illness and its grave prognosis while still president. He died about a year and a half after completing his presidential duties.
Bright's Disease was the general term given to edema and end-stage kidney disease felt to be due to kidney failure. Obviously, it is difficult to impossible to say what the etiology of his symptoms are. He is also described as having a "feeble heart", so it is possible that his edema was in fact secondary to advanced CHF. He also was a reknowned drinker and socializer, so cirrhosis could also be in play, though he was not reported to have other stigmata of liver disease. He is also described as having a chronic struggle with poor appetite late in life, which could also be interpreted as uremia however.
An interesting list of Presidential medical ailments can be found here. Of note, Grover Cleveland (who succeeded Arthur) is listed as having "early nephritis" but I can't any details ( in my admittedly rapid reserach) elsewhere.