Mary,
I think the confusion is that Mathematica "strings" involve text strings as Sander demonstrated. I believe that your "strings" are lists of binary digits (and not Mathematica strings). If this is correct, This should work for you:
In[2]:= list = Range[15]
Out[2]= {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
IntegerDigits converts my list of numbers to 4 digit binary "Strings" in Mathematica -- these are lists of digits
In[3]:= mylist = Map[IntegerDigits[#,2,4]&,list]
Out[3]= {{0,0,0,1},{0,0,1,0},{0,0,1,1},{0,1,0,0},{0,1,0,1},{0,1,1,0},{0,1,1,1},{1,0,0,0},{1,0,0,1},{1,0,1,0},{1,0,1,1},{1,1,0,0},{1,1,0,1},{1,1,1,0},{1,1,1,1}}
Make a function that picks out the lists with a certain number of 1's
In[5]:= SelectHamming[str_List,n_Integer]:=Select[str,Count[#,1]==n&]
In[6]:= SelectHamming[mylist,2]
Out[6]= {{0,0,1,1},{0,1,0,1},{0,1,1,0},{1,0,0,1},{1,0,1,0},{1,1,0,0}}
I hope this helps