Nothing's impossible, but it cannot be done with a straight forward
mailmerge.

If the first name and the last name are in separate fields and you don't
have any

Bill Smith jr.

or

Bill Smith III

Then you could do it in Word by first executing a catalog or directory type
mail merge, then running a macro over that document to create a new document
that would create a new data source consisting of the unique records plus a
single instance of the address where there are multiple addressees with the
name of the addressee changed as to "The Smith Family" as in your example.

The following macro, that I developed for another purpose, does not do what
you want at the present, but it is what I would use as a starting point to
develop one to do what you want:

Dim source As Document, target As Document, scat As Range, tcat As Range
Dim data As Range, stab As Table, ttab As Table
Dim i As Long, j As Long, k As Long, n As Long
Set source = ActiveDocument
Set target = Documents.Add
Set stab = source.Tables(1)
k = stab.Columns.Count
Set ttab = target.Tables.Add(Range:=Selection.Range, numrows:=1,
numcolumns:=k - 1)
Set scat = stab.Cell(1, 1).Range
scat.End = scat.End - 1
ttab.Cell(1, 1).Range = scat
j = ttab.Rows.Count
For i = 1 To stab.Rows.Count
   Set tcat = ttab.Cell(j, 1).Range
   tcat.End = tcat.End - 1
   Set scat = stab.Cell(i, 1).Range
   scat.End = scat.End - 1
   If scat <> tcat Then
       ttab.Rows.Add
       j = ttab.Rows.Count
       ttab.Cell(j, 1).Range = scat
       ttab.Rows.Add
       For n = 2 To k
           Set data = stab.Cell(i, n).Range
           data.End = data.End - 1
           ttab.Cell(ttab.Rows.Count, n - 1).Range = data
       Next n
   Else
       ttab.Rows.Add
       For n = 2 To k
           Set data = stab.Cell(i, n).Range
           data.End = data.End - 1
           ttab.Cell(ttab.Rows.Count, n - 1).Range = data
       Next n
   End If
Next i

Alternatively, I am pretty sure that you could do this in Access.  I am not
sure though whether it would be any simpler.