Clinton understands the details and the players. He naturally symbolizes the achievements of the Abbas-Olmert negotiations, which picked up where the Clinton parameters left off. He is extremely popular in Israel and (through Ehud Barak) very close to the Israeli defense community. He is insanely popular among American Jews.
Clinton also has enormous international prestige and his Global Initiative represents the kind of economic development and political creativity a two-state solution will need. Should Hillary run in 2016, what could be better for her than a peace deal in process. Bill Clinton now has Obama's trust and everybody knows it. He is too big to fail.
The appointment, if it could be made in the next month or so, will be the most subtle way to promise efficient pressure on the Israeli government, discredit Netanyahu's candidacy, and bring a saving prestige to Abbas, without distressing either the Israeli center-right, much of which will anyway support Likud, or American Jews, most of whom indeed support the parameters that bear his name. (See this latest J Street poll by Jim Gerstein, and turn to slide 36.)
President Obama has been advised to keep away from the Israeli-Palestinian issue. This is terrible advice. The status quo in Palestine will bring new and destabilizing violence more dangerous than any Iranian bomb. The young Arab street, especially in Egypt, will never be reconciled to Obama's leadership, or the American model of democratic globalization, so long as Palestinians live under occupation.
Yet, granted, Obama has more immediate priorities. To address the Israel-Palestine issue he needs a public figure to be a lightening rod for all the hard feeling that will come out as the process evolves without forcing Obama himself to get into the weeds or risk his prestige as specific difficulties are leaked. Another George Mitchell will not do. Obama needs something like what Henry Kissinger was for Richard Nixon during the post-1973 disengagement shuttles. That person has been standing by his side for the past month and has unfinished business of his own.