So it doesn't ask for the filename until the end of the program, which is not exactly intuitive. But it does ask for it!
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: ');
# Exercise: make the file name used here customizible:
add_names(new_names, raw_input('Enter a filename: '))