Eddy taught school in a rural district before joining the Northern Pacific Railroad Company as a land examiner in 1881 and 1882. He then moved to Glenwood, Minnesota, and served as clerk of the district court of Pope County from 1884 to 1893.
In 1894, Eddy was elected as the first United States Representative from Minnesota who was a native of the state. He was elected as a Republican to the 54th, 55th, 56th, and 57th congresses, March 4, 1895, until March 3, 1903. While a congressman, Eddy served as chairman of the Committee on Mines and Mining in the 57th congress. He declined to be a candidate for renomination in 1902.
After his terms in Congress, Eddy became editor and owner of the Sauk Centre Herald. He died on January 13, 1929, in Saint Paul, Minnesota, and was interred in Greenwood Cemetery in Sauk Centre.
