Tony was a Cuban-born outfielder who played his entire career (1962-1976) with the Minnesota Twins. He struggled with knee injuries in the last five years of his career -- he was mostly limited to playing DH during this time. Tony was the 1964 AL Rookie of the Year, was selected to eight All Star teams, won one Gold Glove, and had a lifetime .304 batting average. He won three AL batting crowns, led the league in hits five times, and was runner-up for MVP twice (1965 and 1970). Tony was the first DH to hit a home run.
Tony's web site has a page that makes a case for his election to the Hall of Fame. I never saw him play, so I don't know about his HOF qualifications. He had some great stats from 1964-1971. If he gets in, it would be using about the same criteria that got Kirby Puckett and Bruce Sutter in. Here is a good year-to-year rundown on Oliva's career.
Tony was a player-coach in the 1976 season and only batted 123 times.
This card says that his date of birth was 7/20/1940, but his actual DOB was 7/20/1938. He used his brother's birth certificate to get into the country.