I thought I knew the answer to this - but a bit of research showed me I was only half right.
I'd have said that the clock tower of the Palace of Westminster, the UK's Parliamentary building, was the origin. However, it appears I'm wrong - it was the trigger for its popularity but not its origin.
The set of tunes is, though, known as the Westminster Quarters.
Wikipedia helped me out -
This chime is traditionally, though without substantiation, believed
to be a set of variations on the four notes that make up the fifth and
sixth measures of "I know that my Redeemer liveth" from Handel's
Messiah. This is why the chime is also played by the bells of the
so-called 'Red Tower' in Halle, the native town of Handel. It was
written in 1793 for a new clock in St Mary the Great, the University
Church in Cambridge. There is some doubt over exactly who composed it:
Revd Dr Joseph Jowett, Regius Professor of Civil Law, was given the
job, but he was probably assisted by either Dr John Randall (1715–99),
who was the Professor of Music from 1755, or his brilliant
undergraduate pupil, William Crotch (1775-1847).
In the mid-19th century the chime was adopted by the clock tower at
the Palace of Westminster (where Big Ben hangs), whence its fame
spread. It is now possibly the most commonly used chime for striking
Incidental information - the clock is not called Big Ben, the bell that chimes the hours is Big Ben, the clock is just the clock of the Palace of Westminster & actually doesn't really have a name.