In modern-day fiction - particularly in Hollywood movies - Mighty Whitey pops up as the result of creative types trying to appeal to as broad a cross-section of society as possible to get their cash back. One particular version has it so that the sympathetic Author Avatar whitey is not only now the Great White Hope for the non-white Noble Savages, but is very often defending them from other evil whites.
In modern-day fiction, sometimes the Mighty Whitey is there to lead or inspire the Hollywood Natives or bring some aspect of modern technology or knowledge to their aid, something they presumably could not do before he showed up. All this is a setup for the white man to adapt to the Native's ways, thereby making him superior both to the natives and the Europeans back home. Indeed, sometimes the native peoples are shown to be better in some way than European society and the white man begins to despise his old home. Sometimes the foreign societies are shown to be realistic, three-dimensional and actually rather pleasant places to live. Extra points if he woos The Chief's Daughter along the way an unfortunately common variation that perpetuates into present-day media is that she will continue to love our hero even if he is directly responsible for the death of her husband, brother, or father. In cases of the first and second, this trope can heavily overlap with Led by the Outsider.
A common trope in 18th and 19th century adventure fiction, when Europeans were visiting and documenting vast swathes of the world for the first time, Mighty Whitey is usually a displaced white European, who ends up living with native tribespeople and not only learns their ways but also becomes their greatest warrior/leader/representative.