Why wasn't the Big Ben destroyed during the London Blitz?
To my understanding, the Nazis bombed London consecutively night after night for several months. I would assume that a good part of London's buildings were destroyed. Why was the Big Ben not hit?
Your question is perhaps a little tainted with modern thinking.
The bombing of London and its surrounding areas would have been accidental, (and there were many accidents! so it was a miracle certain buildings and areas were missed)
"The war against England is to be restricted to destructive attacks against industry and air force targets which have weak defensive forces. ... The most thorough study of the target concerned, that is vital points of the target, is a pre-requisite for success. It is also stressed that every effort should be made to avoid unnecessary loss of life amongst the civilian population.
—Hermann Göring
However:
On 7 September, 318 bombers from the whole KG 53 supported by eight other Kampfgruppen, flew almost continuous sorties against London, the dock area which was already in flames from earlier daylight attacks. The attack of 7 September 1940 did not entirely step over the line into a clear terror bombing effort since its primary target was the London docks, but there was clearly an assumed hope of terrorizing the London population. Another 250 bomber sorties were flown in the night.
Academy Awards 2010 Oscar Predictions with BIG BEN and FANNY