Answer by taking a quote from the following article:

Jon Stewart was born Jonathan Stuart Leibowitz on November 28, 1962, in New York City, to Marian (nee Laskin), a teacher and later educational consultant, and Donald Leibowitz, a professor of physics at The College of New Jersey and Thomas Edison State College. Stewart's family are Litvak Jews who emigrated to America from Poland, Ukraine and Belarus. One of his grandfathers was born in Manzhouli (now part of Inner Mongolia). He is the second of four sons, with older brother Lawrence and younger brothers Dan and Matthew.

Stewart was an important factor in the unionization of the Comedy Central writers. The Daily Show writers were the first of Comedy Central's writers to be able to join the guild, after which other shows followed.  Stewart supported the 2007-08 Writers Guild of America strike. On The Daily Show episode just before the strike, he sarcastically commented about how Comedy Central had made available all episodes for free on their website, but without advertising, and said, "go support our advertisers". The show went on hiatus when the strike began, as did other late night talk shows. Upon Stewart's return to the show on January 7, 2008, he refused to use the title The Daily Show, stating that The Daily Show was the show made with all of the people responsible for the broadcast, including his writers. During the strike, he referred to his show as A Daily Show with Jon Stewart until the strike ended on February 13, 2008.  Stewart's choice to return to the air did bring criticism that he was undermining the writers of his show. Seth MacFarlane wrote an inside joke into an episode of Family Guy about this, causing Stewart to respond with an hour-long call in which he questioned how MacFarlane could consider himself the "moral arbiter" of Hollywood. Other former writers of The Daily Show such as David Feldman have also indicated that Stewart was anti-union at the time and punished his writers for their decision to unionize.  The Writers Guild Strike of 2007-08 was also responsible for a notable mock feud between Stewart, Stephen Colbert, and Conan O'Brien in early 2008. Without writers to help fuel their banter, the three comedians concocted a crossover/rivalry in order to garner more viewers during the ratings slump. Colbert made the claim that because of "the Colbert bump", he was responsible for Mike Huckabee's success in the 2008 presidential race. O'Brien claimed that he was responsible for Huckabee's success because not only had he mentioned Huckabee on his show, but also that he was responsible for Chuck Norris's success (Norris backed Huckabee). In response, Stewart claimed that he was responsible for the success of O'Brien, since Stewart had featured him on The Jon Stewart Show, and in turn the success of Huckabee. This resulted in a three-part comedic battle between the three pundits, with all three appearing on each other's shows. The feud ended on Late Night with Conan O'Brien with a mock brawl involving the three hosts.

Why did he refuse to use the title?
stating that The Daily Show was the show made with all of the people responsible for the broadcast, including his writers.