Richard Nixon served as Dwight Eisenhower's VP from 1953 to 1961, and was elected as President in 1968 and 1972. Of course he didn't complete 2 full terms as President due to the Watergate scandal which forced his resignation in 1974.
Jefferson, Madison, Jackson, Lincoln, Grant, Cleveland, McKinley, and Nixon all had different vice-presidents during their second terms. Franklin Roosevelt changed veeps for his third term and again...