Shoutout to Jaleesa for the assistance!
def add_names(list_of_names, file):
  """
  Opens and adds a list of names to the end of a file, each on its own line
  """
  # We open a file in 'a' mode, for appending to it.
  names_file = open(file, 'a')
  # For each line in the list, we print that to the file. 
  # This assumes one file per line.
  for name in list_of_names:
    print >> names_file, name
  # Close the file so the changes are visible.
  names_file.close()
# Exercise: make new_names customizible:
new_names = input('Enter a list of names: ')
print ('Name written to file')
# Exercise: make the file name used here customizible:
add_names(new_names, input('Name file: '))
print ('File named')