I must confessed I missed this on Comment is Free until Conservative Home helpfully highlighted it this morning, but this is an interesting question which deserves a slightly better answer than the one which Peter Franklin supplies.
Franklin says yes, if Gordon Brown were to be dragged down by Labour infighting and Blairite attempts to find an alternative from within their own ranks fail. I say: still extremely unlikely, even in the event of these other conditions being fulfilled.
I know Beckett reasonably well, as it happens, from my days as Political Correspondent of the Derby Evening Telegraph, her local paper. I have never found her to be anything less than extremely courteous, and for the most part, her career has exhibited that much-prized attribute, of being a "safe pair of hands."
Nevertheless, she has made two crucial errors in the course of her time in frontline politics which I think still, to an extent, define her in terms of her political positioning and which the Tories - and their friends in the media - would relentlessly exploit if she ever assumed the top job.
The second of these I have referred to in my post on John Prescott below. Beckett could easily have become Labour leader in 1994, but threw away her chance by failing to give sufficient backing to the modernising cause.
Much earlier than that, in 1981, she launched a bitter attack on soft-left MPs who had abstained in that year's deputy leadership contest, thereby allowing Denis Healey narrowly to hold off the challenge of Tony Benn.
It was this action that originally earned her the nickname "Stalin's Grandma," an epithet which was also applied to Jo Richardson and has since been appropriated and adapted by a well-known journalist blogger.
They say a week is a long time in politics, but even now, a quarter of a century on, the idea of a 63-year-old former Bennite up against someone with the wide electoral appeal of David Cameron is surely not one that the Labour Party would be wise to contemplate.
My position remains that Brown will comfortably win the Labour leadership, unless the economy suddenly goes belly-up or forthcoming attempts to link him with the cash for honours affair succeed.
Failing that, the job will go to Alan Johnson, or possibly to John Reid, who if nothing else in his brief tenure of the Home Office has demonstrated his extreme political toughness.
But if by then the party is too divided to accept either a Brownite or a Blairite as leader, it won't be Beckett who comes through the middle as the compromise candidate associated with neither side, but the man whose old dad she once so enthusiastically championed.
Step forward, the Rt Hon Hilary Benn.