The expected move of Hillary Clinton to Foggy Bottom will set off an intriguing parlor game among New York State Democrats. Whom will Governor Paterson appoint to succeed Clinton? And how will it affect the gubernatorial and senate races in New York in 2010?

New York Attorney General Andrew Cuomo has long been seen as a likely primary opponent to Paterson in 2010, and given his famous name and fundraising base, he’s likely to offer a stiff challenge. Paterson will surely be tempted to dispatch Cuomo to Washington — if he’s willing to go. Recent reports say Cuomo would turn the job down, but his attitude might change once the discussion moves past hypotheticals.

On the other hand, the New York State chapter of NOW wants a woman to succeed Clinton. That could boost the chance that New York City Council Speaker Christine Quinn could become the first openly lesbian member of the U.S. Senate. And if all that speculation weren’t enough, the New York Post reports that Paterson and Senator Chuck Schumer may already have a deal to appoint a relative nobody to the spot. It seems that Schumer’s priority is that he get recognized as New York’s senior Senator, and Andrew Cuomo would be the only contender for the Senate seat to threaten that status. Schumer apparently wants a less high-profile junior Senator — someone like Buffalo Representative Brian Higgins, or Congresswoman Kirsten Gillibrand.

Whoever ultimately succeeds Clinton, both the successor and Paterson will be up for re-election in 2010. Waiting in the wings are New York Mayor Michael Bloomberg (whose popularity has dipped since he announced his plan to seek a third term), and former Mayor Rudy Giuliani. A recent Marist poll shows that even as Paterson’s popularity falls, he still leads both Bloomberg and Giuliani. But Giuliani may have an ace-in-the-hole if he runs for Goveror in 2010: he will be the only one of the three candidates who won’t be responsible for eliminating massive budget deficits in the next two years. Both Bloomberg and Paterson are looking at tax increases and spending cuts to address the deteriorating fiscal state of New York City and New York State, respectively. Neither one may be sporting a shining resume in 2010, and voters may be ready for new leadership.