In the US, there had been a tradition of the term family name meaning the surname of the husband in the family. While this was a prevalent trend, there have always been families in which there was no husband actively present, (no marriage, separation, divorce, death), and circumstances where members of the family used other surnames (both spouses keeping their own names, hyphenated surnames, remarriage). There are numerous patterns of people in families changing their surname (upon marriage, divorce, remarriage, adoption, emancipation).
Many forms used by governmental agencies and businesses include the term family name, but others use the term surname. However, in most states in the US, an individual is free to use any name, given (first name) or surname he or she wishes, often without formal governmental involvement, so long as it isn't for a fraudulent or illegal purpose. Usually the most import issue is consistency of name or a clearly documented pattern of name change to avoid confusion.
In general, there are no formal rules in the US about requiring people to either keep or change their surname at marriage, or how children must be given surnames. Currently there widely divergent patterns of how people use surnames, even though it is still fairly common for women who marry men to adopt the male spouses surname and to give children of married couples the surname of the father. Again, there is wide variability.
As such, there really is no such thing as a standard name for a family. While it is common to refer to a family who all share the same surname as the Smith Family, this si a convention that makes no sense when members do not share a surname (unless they choose to refer to themselves in that fashion).
Where you know members of the family do not have a universal surname, and they have not labeled themselves with a common family designator, it probably is best for you not to make one up. Just as the mix of items in a salad may not fit a common single class label, the family members may not have a single word that is common to refer to them.
If you are writing within a document, consider limiting your labeling to your family.