I very much doubt that your great grandfather was a Bow Street Runner. They were founded in 1749 so, if your relative was a member of the initial group of eight men, he'd probably be something like your great great great great great great great great grandfather. Even if your relative was in the Bow Street Runners when they were disbanded in 1838, there ought to be several more 'great's in the relationship than your question implies.
Check the sums: If you were born in, say, 1960 and your father was 25 years old at the time, that means your father was born around 1935. If his father (your grandfather) was 25 years old at the time, that means that he was born around 1910. If his father (your great grandfather) was 25 years old at the time, he'd have been born around 1885, which was far too late for membership of the Runners.
If any records of the membership of the Runners have survived the ravages of time (and the attentions of Hitler's Luftwaffe), the Museum of London will probably hold them (or know where they're located). Telephone them on 020 7814 5750. (That number goes directly to the department which deals with London's history from 1700 onwards). Alternatively, e-mail the department:
[email protected]
Chris