He attended New York University Law School and came west to look after business interests of the wealthy Stokes family.
He was the 12th Governor of Nevada and a United States Senator. He joined the Nevada Bar in 1898. Oddie moved to and made his fortune in the 1900 silver boom in Tonopah, becoming manager of the Tonopah Mining Company.
He was the Nye County District Attorney from 1900 to 1902.
On March 17, 1911 he signed the city charter for Las Vegas. During his tenure, women got the right to vote, a state motor vehicle law was sanctioned, mining safety legislation was endorsed, and there were improvements to workmen"s compensation benefits.
He was a senator from 1921 to 1933, losing his bid for a third term to Pat McCarran in 1932. Oddie died on February 17, 1950 in San Francisco, California, at the age of 79.
He is interred at Lone Mountain Cemetery, Carson City, Nevada.
Mount Oddie near Tonopah is named after him, as is Oddie Boulevard in Reno and Sparks.
He was a member of the Republican Party. He was a member of the Nevada State Senate from 1903 to 1906.