WHENEVER a mayor runs for reelection on the ground that he has "learned, grown and changed," you know he has made some big mistakes, and that is true even when, as in the case of Mayor Jane Byrne of Chicago, the he is a she. This is not to say that her cause in the decisive Democratic primary next February is lost. Other mayors have come back from bigger deficits and won new terms. John Lindsay ran for reelection in 1969 on the ground that he was "not perfect," and voters, confronted with the arguably less perfect alternative of Mario Procaccino, decided that was good enough. In 1961, Robert Wagner ran for mayor of New York as the reform opponent of the Democratic machines that had warmly endorsed him in the two previous elections. He won handily.

Anyway, the old rules went by the boards when Mrs. Byrne won the Democratic primary in 1979. She beat Michael Bilandic, the incumbent, a longtime resident of the same 11th Ward that had provided all of Chicago's mayors since 1933; Mr. Bilandic made the mistakes of moving to the Gold Coast and failing to clear Chicago's streets of snow. The machine was supposed to dominate this all-important election, originally scheduled in the dead of winter on the theory that only the machine can get the vote out in the darkness and bitter cold of a Chicago February; but the machine lost. Now Mayor Byrne, possessed of the power of appointment, has the endorsement of the Democratic County Committee, over her opponents, Cook County State's Attorney Richard M. Daley (yes, he's the late mayor's son) and South Side Rep. Harold Washington.

If you had told anyone 10 years ago that the mayor of Chicago today would wear dresses, he would have thought you were making a tasteless joke about Mayor Daley. But Jane Byrne, whatever happens to her in the primary, whatever Chicagoans think of her performance, has indisputably been mayor. She has changed her mind and made mistakes; she has had one of the lowest job ratings of American officials. But each day, she has left home -- her Gold Coast apartment or, for a while, the Cabrini-Green housing project -- gritty and determined to give as good as she got. She has faced opposition as hard-nosed as any American politician gets and a press as critical. Tough and unsentimental, she changed Americans' ideas about the role of women in politics and government: she has shown that a woman can run Carl Sandburg's Hog-butcher for the World, City of the Big Shoulders.