Elie Wiesel will leave for the Soviet Union today, a week after winning the Nobel Peace Prize, to try to persuade Soviet leader Mikhail S. Gorbachev to let Jews and dissidents leave the country. "I would like to . . . see if it's possible to find a common language simply because we speak on behalf of suffering," Wiesel told a news conference at Boston University, where he teaches. He said that the purpose of his weeklong trip to Moscow, planned before he learned that he had won the Nobel, is to invite the Soviet Union to a Washington conference on non-Jewish victims of the Holocaust.