Choose random line of input while storing only one line at a time and an integer with uniform probability.

random_input_line.py

# only variables I'm allowed to store/modify
i = 0
stored_string = "No input given!"

# we'll need to be able to make random choices
import random

# solution
while True:
  try:
    i += 1 # count the place of this string/person in line
    if 1 == random.randint(1,i): 
      # make a decision to keep it with probability 1/i
      # note for first person, i=1, so this passes
      stored_string = raw_input()
    else: 
      # otherwise ignore it
      raw_input()
  except EOFError: # ran out of people in line, we're done
    break

# Claim: the line stored had equal probability of being chosen with any other line
print "Result:\n%s" % stored_string