One of the most underdeveloped political dynasties is IMO, the Roosevelts. For putting out both T.R. and F.D.R. and having a number of other political figures in both the Oyster Bay (Republican) and Hyde Park (Democratic) branches of the family.
If you want an Oyster Bay dynasty to follow Teddy, look no further than T.R.'s son, Theodore Roosevelt Jr. Teddy Roosevelt Jr. actually was the GOP nominee for Governor of New York in 1924, but thanks to vigorous campaigning from his cousins Eleanor and Franklin for Smith and against him (thanks largely to his involvement in the Tea Pot Dome mess while Secretary of the Navy), he went down by an extremely small margin to Smith. Have butterflies shift a few votes around, and in 1925, you've got Governor Theodore Roosevelt, Jr.
Junior will probably get re-elected in 1926 and 1928, but probably will either retire in 1930 or get booted out because of the Depression. When Hoover gets thrown out in '32 by whomever conservative Democrat is nominated (likely not F.D.R., with T.R. occupying his OTL time in Albany), while T.R. plans his own run for President down the line. Conservadem wins re-election in 1936, but by 1940, the field is ripe for a challenge to the economic orthodoxy. T.R. is elected President by a hefty margin, and begins stepping up U.S. preparedness efforts for war in Europe.
Other Oyster Bay Roosevelts that could possibly get elected to some sort of office are rather slim. There's Archibald Roosevelt, but then again, there's the tiny fact that he was an extreme right-winger. And I'm not exaggerating, either. He was a member of the John Birch Society.
On the other hand, the Hyde Park Roosevelts are a plenty with possible ATL presidents.
James Roosevelt, the eldest of the Hyde Park branch, was actually a U.S. Representative from California from 1955 to 1965. Prior to that, he attempted to run for Governor of California in 1950, but lost to Earl Warren, and attempted a primary challenge to Sam Yorty for Mayor of Los Angeles, but again, came up short. The easiest POD I can think of here is somehow have Roosevelt manage to knock Sam Yorty out of the primary and win the general election. He serves as mayor of Los Angeles until the end of the Reagan governorship, then moves to run for the governorship himself, which he's elected to in 1974. Of course, you can have him make a bid for the White House in '76 (when he'd be nearly seventy years old). It'd be a long shot, but older men have been elected from the California Governorship to the Presidency, after all.
Franklin Roosevelt Jr., is the second most politically astute of the Hyde Park Roosevelts. He served in Congress from 1949 until 1955, and was nominated for Attorney General of New York in 1954, though he would ultimately lose the election to that post. Let's say butterflies have him become Attorney General in '55, serve until '63, and then head for the Governorship against Nelson Rockefeller. In a battle between the Roosevelts and the Rockefellers, let's give the edge to the Roosevelt, who becomes Governor of New York in 1963. Re-elected over Rockefeller again in 1966, but at a much smaller margin, Roosevelt sets his sights on the White House, but decides against a run in 1968, which looks like it's going to be sheer chaos. After winning re-election in 1970, he makes his bid for the Democratic nomination in '72, and is granted the V.P. spot on a ticket headed by Henry Jackson. Jackson goes down, but Roosevelt presses forward, and is nominated by the Democrats in 1976 as their candidate for President...