Pick a random line from a file / stream without knowing how many lines there are to choose from in one pass without storing the lines that have been seen.

  perl -e 'while(<>){$x=$_ if rand()<=(1/$.)}print $x'

For each line, pick that line as your random line if a random number (0<=n<1) is less than the reciprocal of the number of lines read so far ($.).

It hits my elegant bone. Only one line... rand < 1/1, pick it. Two lines, same as one, but the second line has a 1/2 change of replacing line one. Third line same as before but gets a 1/3 chance of taking the place of whichever line has survived the first two picks. At the end... you have your random line.