Yellen would also take over the Fed at a delicate time for China, the world's No. 2 economy after the United States, and other developing nations.
The International Monetary Fund, citing slower growth in China, India and Brazil, this week reduced its forecast for global economic growth to 2.9 percent this year and 3.6 percent in 2014. Both are 0.2 percentage point weaker than the IMF's previous forecast in July.
Investors have been pulling money out of developing markets, partly to take advantage of rising interest rates in the United States. The Fed might be called upon to help calm worldwide financial volatility.
"The role of the Fed chair is so critical," says Greg McBride, senior financial analyst at Bankrate.com. "We're not just talking about the U.S. economy. We're talking about the global economy."
Yellen will also have to establish herself as chair at a time when the Fed is experiencing unusual turnover. When Bernanke leaves Jan. 31, but there could be up to four vacancies that Obama would need to fill on the seven-member Fed board.
One board member, Elizabeth Duke, left in August. Another, Sarah Bloom Raskin, has been nominated by Obama to become deputy Treasury secretary. The term of a third, Jerome Powell, will be up Jan. 31, though he can remain on the board until a successor is confirmed by the Senate.
And Sandra Pianalto, president of the Federal Reserve Bank of Cleveland, has announced that she will leave early next year.
Most analysts say they're confident Yellen can handle the many challenges.
She's "a good choice," McBride says. "You want your Fed chairperson to be cool, calm and collected and a steady hand. Yellen is certainly that."