The year 1057 seemed to have been a pivotal one for the history of England, and the events leading up to the Norman invasion may very well have begun ten years before 1066. The legend of Godiva places her ride about 1057, the same time that Leofric of Mercia died, as did Aedward the Exile. Tostig had secured the Earldom of Northumbria by 1057 as well. I have read nothing that has linked the house of Godwin to either the death of Aedward the Exile or that of Leofric. Nevertheless, it is interesting that the two deaths happened about the same time.
Harold had struggled with the son of Leofric in the years following the old Earl's death, and he was exiled at one point, although reinstated shortly thereafter. Aelfgar proved to be a troublesome young earl, and he may very well have died during one of his raids with his Welsh allies.