There is debate over the origin of the name Big Ben. The name was initially applied to the Great Bell. It may have been named after Sir Benjamin Hall, the overseer of the installation of the Great Bell. Some think it was the English heavyweight boxing champion Benjamin Caunt.
The construction of the tower began in 1843. Clock maker Edward John Dent was hired to design the clock in February of 1852. Dent past away a year later and was replaced by his stepson, Fredrick Dent.
During the period of 1856-1857, the Great Bell is cast by John Warner & Sons at Stockton on The Tees on August the 6th 1856. The bell developed a crack while being tested prior to installation. Therefore, a replacement bell is to be cast. The new contract was given to White Chapel Bell Foundry. A very detailed early history of the Great Bell can be found on the White Chapel Bell Foundry website.
On April the 10th 1858 a replacement bell being 2.5 tons lighter than the first bell is cast. It took two weeks for the new bell to cool. It was transported to Westminster on a carriage drawn by 16 horses. Big Ben chimes for the first time in 1859. In September of 1859 the Big Ben bell cracked. For the next several years the bell was out of service and the hour was struck by the largest of the quarter bells. The culprit for crack in the Big Ben bell was the hammer. It was too large and cracked the bell. A lighter hammer was installed. Big Ben was turned to present the hammer away from the cracked portion of the bell. This is the same bell you hear today with the crack.

Other historical notes:
1923 New Years Eve chimes were broadcast by BBC radio.
For two years during World War I, the bells were silent and the clock faces were not illuminated at night. This was to avoid detection and attack by the German Zeppelins.
During World War II the bells continued to ring, but the dial faces were not illuminated at night. This was to avoid detection of German bomber pilots during the Blitz.

Accidental stoppages:
June 3-4, 1941. The clock stopped from 10:13 PM until the following morning due to a repairman repairing damage from an air raid dropped a hammer into the movement works.
In 1949 the clock slowed by 4-5 minutes from a flock of starlings who were perched on the minute hand.
January 13th, 1955, the clock stopped at 3:24 AM due to snow and ice forming on the east dials. After this occurrence, small heaters were placed on the dials facing eastward to avoid freezing.
On December 31, 1962 in spite of the measures to prevent freezing, the clock slowed due to heavy snow on the hands. Due to this the new year of 1963 was late to chime by nine minutes.
The clock was purposely stopped:
On January 30, 1965 the bells were silent during the funeral for former Prime Minister Winston Churchill.
1976 A completely unanticipated event occurred which almost caused the complete destruction of the clock. After more than 115 years of use, at 3:45am on the 5th of August 1976 the clock started to chime. The metal fatigue in the shaft connecting the chiming train to its fly fan caused the shaft to break. Without the braking effect of the speed governing fly, the chiming mechanism propelled by the 1.25 ton weight in the shaft, dramatically increased its speed of rotation. This led to the total destruction of the chiming mechanism, with various components and fragments of others being scattered around the clock room. Some pieces were flung at the ceiling with sufficient force to penetrate to the room above. The iron frame was fractured and collapsed onto the winding motor below. Flying debris also caused damage to the going time and striking trains.
It was necessary for the chiming train to be totally reconstructed. The enormity of this task meant that options, such as replacement with an electric motor were considered. The reconstruction took almost one year to complete.
Big Ben has been silenced again as of August 21, 2017 for major repairs and renovations. It is expected to resume full service in 2021.
Visiting Big Ben and Elizabeth Tower is possible:
If guests fit a strict criteria and have contacted their MP or a member of the House of Lords to request a visit. Tours tend to be sold out for up to 6 months in advance so be prepared to wait a while. More information can be found at their website.