New Bridewell was a prison built c. 1617 next to Clerkenwell Bridewell, which was a prison located in Clerkenwell http://www.pepysdiary.com/encyclopedia/3395/ named 'Bridewell' after the Bridewell Palace, which during the 16th century had become one of the City of London's most important prisons. http://en.wikipedia.org/wiki/Clerkenwell_Bridewell
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From The Book of Days (1869)
Adjoining to St. Bride's Churchyard, Fleet-street, is an ancient well dedicated to the saint, and commonly called Bride's Well. A palace erected near by took the name of Bridewell. This being given by Edward VI to the city of London as a workhouse for the poor and a house of correction, the name became associated in the popular mind with houses having the same purpose in view. Hence it has arisen that the pure and innocent Bridget--the first of Irish nuns--is now inextricably connected in our ordinary national parlance with a class of beings of the most opposite description.
For the new prison see the upper left-hand corner of this 1746 map
Chart showing the number of references in each month of the diary’s entries.