Not everyone in a family may have used the same spelling of a last name. In this immigrant family, the mother used the spelling of Müller and the son used Miller. The difference in this case is small, but for names that were more difficult to spell or pronounce, the change could have been more significant, making it difficult for the researcher to immediately see the possible connection between the two names.
Non-immigrant families occasionally change their names as well–often due to family squabbles, the desire to mask family origins, etc.
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