How does one address a blended family in which the members have different surnames?
Solution 1:
On the outside, it's perfectly acceptable to address the envelope to the head of the family alone. For comparison, when you write to your bank, it's traditional to address the letter to "The Manager", even though he/she might not be the one acting on your correspondence. You could also put "Ms. Smith and family" on the envelope.
On the inside of the card is a different matter. You probably have more space to write; keeping things short is not as important. It's also potentially a more informal setting. (Unless you were sending a condolences card, which would usually be formal unless you were very close to them.)
In Australia, where I'm from, using the phrase "the Jones family" is very unusual as a form of address, though I gather it must be a more common usage in the USA. We usually address the inside of the card using given names (starting with the adults), e.g. "Dear Helen, Steven, Ben and Sarah". But then Australians these days use first names for business relations and even superiors, which might not be the case where you come from. It's also a common usage here to address the people you know by name, adding and family to include the others, e.g. "Dear Helen, Steven, and family" (in this example, Helen and Steven are the adults whilst Ben and Sarah are the children).
You say that the family is unknown to you, do you mean that they are all strangers? Or that you have met the mother but not the other family members?
Is there some social connection? For example, are the children connected to your own children through school, sport, etc.? In that case, you may want to list the children by their given names, rather than include them in "and family". Unlike children addressing adults, it's rarely wrong for an adult to address a child by their first name. It might look weird to write "Dear Ms. Smith, Bob, Ben, Amy, and family", you would have to judge whether you like that style.
If you have to stay formal, then "Dear Ms. Smith and family" might be your only option apart from "the Smith-Martin-Jones family".
Solution 2:
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.