I'm a little late to this thread, but one of the lines of descent in my wife's tree is the Stedman family. They are sometimes spelled Steadman in legitimate documents such as marriage bonds, etc. However, I know for certain the surname is Stedman, because the family has been traced back to England in the fifteenth century.
Stedman is the predominant spelling. But Steadman is not infrequent, although I don't recall seeing any gravestones with that spelling.
Another family is the Seagraves family. From which derives Segrove, Segrave, Seagrave, and even the NC town Seagrove. In that particular case, I have no idea which name is the correct one. I suspect it's Segrove, but I can't prove that.
In some cases, you will never know what the correct name is. You can even find family lines where the name changes over time and the new spelling was generally adopted by all the descendants.
It's an irreducible conundrum.