John Abraham Roosevelt was born January 30th, 1940, to Theodore Roosevelt Jr., who decided against naming his son after his father. He followed family tradition by going to Harvard, and becoming a lawyer after attending law school. He was elected to the New York Senate in 1970, and again in 1972. He was elected continuously, and by the time 1978 rolled around he had gained considerable fame in the state, going on to win the election for Governor. He served the state well, and in 1984 was the Vice Presidential Candidate. There was speculation that we he would run himself in 1988, but he declined, although the seed had been planted in his mind. In 1994, he would become the frontrunner after the complete and utter collapse of the economy thanks to the stupidity of "Reaganomics" which had ruined America's economic system. Roosevelt ran on a left-wing Democratic platform, and with the backlash against right-wing ideas he became the frontrunner, and eventually presidential candidate. In a rather unexpected move, Roosevelt picked Ross Perot as his running mate, giving his ratings quite a boost with independents, and, after a few memorable speeches, the Democratic voting base. He would win the election of 1992, and throughout his first term enact things such as Universal Health Care, and a reform of the Social Security system. He easily won a second term, and towards the end of his term, in 1998, the White House and World Trade Center were both attack by terrorists, the former being partially destroyed. His excellent handling of the events led to an enormous popularity boost, and at the Democratic Convention he was nominated to run for a third term*. He picked a different Vice Presidential candidate for this election: Al Gore. He successfully won, becoming the second President to obtain three terms.
*The 22nd amendment having been repealed during the height of the Reagan years, with Roosevelt being the first to take advantage.
It's 20 years earlier than you wanted, but I think it fits better this way. Also, his fourth term would finish up around 2008.