According to 'English Language Centres', “draw a
blank” originated in Tudor England when Queen Elizabeth I set up
the first national lottery in 1567.
For this lottery, there were two pots. One pot contained slips
of paper with the names of all of the participants. The other
pot contained the same number of slips, only some bits of paper
had prizes written on them while others were blank. One slip of
paper was pulled from each pot at the exact same time and if the
person matched with a prize, then they would win the prize.
However, if the person’s name was drawn with a piece of paper
with no writing, then they would not win anything. They were
unsuccessful and “drew a blank.”