You need a personality-based POD, because beyond a bit of speculation in 2005, and again in 2008 during McCain's veep picks, Rice has never expressed even the slightest interest in running for any elected office. She's a brilliant academic and policy wonk, not a politician, and I've never seen much evidence she has the slightest interest in domestic policy.
I remember vaguely that she was a foreign policy adviser to Gary Hart in the 1984 presidential campaign. By this point, though, her very hawkish views were set. But if Hart wins the nomination but loses the general election, she may gain a lot of exposure. Neoconservatives weren't as unwelcome in the Democratic Party of the 1980s as they are now -- think Daniel Patrick Moynihan, the young Joe Lieberman, and even Scoop Jackson only died in 1983. So if Hart wins the nomination in 1984 Rice could be regarded as a Democratic foreign policy guru, surreptitiously changing her registration back to Democratic (after changing it to Republican in 1982).
So she becomes an influential figure in the emerging Democratic Leadership Council, advocating Democrats reject the 'peace-first', 'blame America first' (not my words) line struck by the party since Vietnam. She probably gets a permanent editorial post at The New Republic -- maybe she even replaces Hendrik Hertzberg as editor in 1985, which wouldn't be the first time the TNR editor has come completely out of the blue. She signs up as a foreign policy adviser to the very young Al Gore (then running as the business-friendly conservative southern candidate) in the 1988 presidential primaries. He loses, but her star rises still further. And through the DLC, she makes connections with an ambitious Southern pol named Bill Clinton. She becomes an influential adviser to his campaign, then his first Ambassador to the UN (replacing Madeleine Albright), then his Secretary of State in 1997. So in 2001 she's roughly where she is now in OTL.
What's changed, here? Well, now she's spent 16 years deeply immersed in the tricky business of electoral politics. She has contacts in the Democratic establishment. She's been an influential figure in running three campaigns, and if you can withstand three campaigns you're probably hooked for life. Through editing TNR she's become a major figure in domestic policy debates as well, the consumnate 'New Democrat'. She's a very conservative Democrat, hawkish on foreign policy and libertarian on economic policy, but a Democrat nonetheless.
Al Gore may pick her as his running mate in 2000 -- if he wins, she's assured a nomination at some point. If he doesn't pick her, or he still loses, she can run instead of Hillary for the New York senate seat in 2000 (having edited TNR, she has connections in the state, and Moynihan would vastly prefer her to Hillary), or she can run in 2000 in Virginia (a trickier bet, but she'll play well in the Northern Virginia suburbs owing to her DC connections), or even run for Governor of California in the 2003 recall (instead of Cruz Bustamante as the Democratic candidate). And these will all be good launchpads for a run.