Who will replace Hillary Clinton as New York senator?

[cnn-photo-caption image= http://i2.cdn.turner.com/cnn/2008/images/11/21/art.clinton.gi.jpg caption=" Clinton's looming exit to be Obama's secretary of state means a scramble for her old job."]
(CNN) - The scramble to replace Hillary Clinton on Capitol Hill was well underway by the time President-elect Obama officially nominated her Monday morning. Among those mentioned to to take her seat as New York's junior senator: her husband, former President Bill Clinton.

Sen. Clinton said at the Monday event announcing her nomination that she wanted to "thank my fellow New Yorkers who have, for eight years, given me the joy of a job I love with the opportunity to work on issues I care deeply, in a state that I cherish."

She added that "leaving the Senate is very difficult for me."

The task of choosing a successor to Clinton will be just as tough.

That job falls to David Paterson, New York's Democratic governor. Whomever he picks would serve for two years, before a special election would be held in November 2010 to decide who fills out the last two years of Clinton's term.

Paterson — who has already taken himself out of the running - has a strong bench to choose from.

There are a number of possible contenders, including at least eight members of New York's delegation in the House of Representatives, New York State Attorney General Andrew Cuomo, Buffalo Mayor Byron Brown, Bronx Borough President Adolfo Carrion, Nassau County Executive Thomas Suozzi, Caroline Kennedy, and her cousin, Robert Kennedy Junior.

"This is not an election. This is not a campaign. It's a constituency of one: David Paterson. It's all about what the governor wants to do," says non partisan political analyst Stuart Rothenberg, editor and publisher of The Rothenberg Political Report. "Paterson has said he would prefer someone from upstate New York, or a woman or an Hispanic candidate."

As for some of the more unconventional picks, Rothenberg says that Paterson could “try to make a splash with a big name like Robert Kennedy Jr." or a "quirky interesting pick" like Caroline Kennedy, who's not a politician.

But some are suggesting that the former President replace his wife.

In an op-ed last week in the Washington Post, journalist and author Karl Meyer and his wife urged Governor Patterson to "send Bill Clinton to the Senate."

If that happened, Clinton would become only the second man to be elected president, then serve on Capitol Hill. President John Quincy Adams lost his re-election bid in 1828. Two years later, he returned to Washington after winning election as a congressman from his home state of Massachusetts, and served in the House of Representatives until his death in 1848.

President Andrew Johnson, who served out the remainder of President Abraham Lincoln's term, also served briefly in the Senate in 1875 - just seven years after Congress voted to impeach him.

Bill Clinton would bring gravitas to the job - and he definitely knows his way around Washington - but some Democratic strategists who used to work for the former president think it’s unlikely he’d want to go from dealing with weighty world issues as the former leader of the free world to worrying over dairy prices and transportation appropriations as the junior senator from his adopted home state.

Clinton’s office told CNN the former president is deferring all questions on his wife’s Senate replacement to Paterson. The New York governor said Monday that he is holding discussions on the question, but that he expects to reveal his pick when Hillary Clinton resigns, and the vacancy officially occurs.