Charles S. Robb, who was elected last week as the first Democratic governor of Virginia in a dozen years, is everywhere identified as Lyndon B. Johnson's son-in-law, as if that gave a clue to his political direction or mission. Anyone who knows young Robb's conservatism will know he is not about to lead a rebirth of the Great Society.

If he has any larger role in life than bringing Lady Bird's grandchildren to the governor's mansion in Richmond, it may be to help keep Ronald Reagan from becoming the Republicans' FDR. That may seem a bit grandiose an assignment for the former Marine and White House military aide whose charming manners won, first, the president's daughter and then the voters of Virginia.

But the more you look at the political scene, the more it appears that the toughest obstacle to the emergence of the always emerging "Republican majority" is the stubborn persistence of Robb's strain of conservative southern Democrat.

Every time it looks as if the GOP is about to end the dominance the Democrats achieved under FDR a half-century ago, some good old boys come along to foil the plot. Under Reagan, the Republicans have had the best chance they've ever had, but it begins to look as if this time, too, the Republicans will be stopped at the Mason- Dixon line.

Reagan carried every southern state except Georgia against Georgian Jimmy Carter. For the first time in history, Republicans won the majority of contested Senate races in the region. Yet the Democrats just plain refuse to roll over and play dead.

In both Carolinas, in Florida and Georgia, in Mississippi, Alabama, Oklahoma and Kentucky, and now in Virginia, Democrats hold the governorships and most of the other key state offices as well. Unless and until the GOP can put feet on its heady top-of- the-ticket wins in Dixie by building its dominance into the statehouses and courthouses, it will lack the national base to become a majority party.

What is significant about the Robb victory is that Virginia, more than any other state of the old Confederacy, appeared ready to grant Republicans that kind of hegemony. It had last supported a Democrat for president in 1964, for governor in 1965 and for senator in 1966. Nine of its 10 members in the House of Representatives are Republicans. And both among the young suburbanites and the aging gentry and business leaders, being Republican has been the socially acceptable thing.

Enter Chuck Robb, a conservative untinged by racism, and suddenly the Democrats recreate a coalition of blacks and whites, of liberals, moderates and conservatives, and they sweep the three elective state offices.

If this were unique, you could attribute it to the campaign shortcomings of the 1981 Virginia Republican ticket. But the same thing has happened in state after state across the South, as Republicans gain the governorship for a term or two, and then are swept aside by the resurgent, biracial Democratic coalition.

A big reason for this is that Republican campaigns and Republican administrations so often ignore or reject the interests of blacks as to virtually drive black voters into alliance even with conservative Democrats like Robb.

Another big reason is that southern Democrats of the new era have become increasingly skilled at mixing careful attention to the business interests of their states with Populist care for at least some "people's issues."

By chance, I have had conversations with three such southern Democratic senators in the past week, Ernest F. (Fritz) Hollings of South Carolina, J. Bennett Johnston Jr. of Louisiana, and David L. Boren of Oklahoma. Far from feeling uncomfortable in the Reagan era, all three say they are more secure in their states and in their party than they were during most of the 1970s.

With minor variations, they tell the same story: they were frustrated by the issues that seemed to dominate the national Democratic Party in the early 1970s, as the causes of the day took the party far from the course their constituents at home could understand or accept. Later, in the presidency of Jimmy Carter, they suffered the embarrassment of seeing his ineptitude ascribed to his southernness, a charge that really rankled.

But now they see the national party "returning to its senses," as Hollings said, and openings emerging for them and their likes in the leadership of the Senate and the party. Boren has the charter to frame a 1982 political "consensus" for the Senate Democratic caucus. Johnston will tell you that a southerner like Reubin Askew could be on the 1984 national ticket, because Carter "didn't really poison the well."

There isn't a hint of their walking out of the Democratic Party or being easy victims for Republican challengers at home. And they look on Robb's win as a sign that Republicanism is not necessarily the wave of the future in the South.