How about moniker for the noun?
And I'd be inclined to invent monikerize for the verb.
(I also like appelation, but I don't have the right to up-vote it yet.)
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".
Best Answer
The name after the first and middle names is usually a surname or family name.
These are often called double-barrelled names or double-barrelled surnames.
In much of the west, people usually have at least a first name and a surname. Sometimes they will have one or more middle names.
(There are many cultural exceptions: Hungary has the family name first and the given name last, Portugal with one or two given names and two family names, Iceland is patronymic and sometimes matronymic: the child's surname comes from one of their parent's given name.)
Usually people will be referred to by friends and family by the first name, but occasionally people will go by one of their middle names. This name is used to distinguish them from other people in the same family or with the same surname.
A surname is also known as family name.
The first name is sometimes called a forename and sometimes called a Christian name, whether the person was christened or not. A middle name can also be called these (although first name is usually restricted to the very first one), and in the plural they can refer to the first and middle names together.
A nickname is a shortened version of a given name, or can be a fun or even cruel name to refer to someone more familiarly.
Others end in -onym. For example, a demonym is a resident of a place (eg Mancunian), a fake name or pen name is a pseudonym.
The list goes on and onnic. You can start listing groups of names (eg different types of given names such as Hebrew names, Germanc names, Celtic names), or names of things.