One Walter C. Pepys, a genealogist, did some research on this. This quote comes from the introduction to the Standard Ebooks edition linked in another comment:

One Walter C. Pepys, a genealogist, did some research on this. This quote comes from the introduction to the Standard Ebooks edition linked in another comment: “The accepted spelling of the name ‘Pepys’ was adopted generally about the end of the seventeenth century, though it occurs many years before that time. There have been numerous ways of pronouncing the name, as ‘Peps,’ ‘Peeps,’ and ‘Peppis.’ The Diarist undoubtedly pronounced it ‘Peeps,’ and the lineal descendants of his sister Paulina, the family of ‘Pepys Cockerell’ pronounce it so to this day. The other branches of the family all pronounce it as ‘Peppis,’ and I am led to be satisfied that the latter pronunciation is correct by the two facts that in the earliest known writing it is spelt ‘Pepis,’ and that the French form of the name is ‘Pepy.’ ” The English sure seem to have a fondness for weird pronunciations. Not long ago I was informed that P. G. Wodehouse’s surname is actually pronounced “Woodhouse,” and even more recently I learned that “Ralph Fiennes” rhymes with “safe signs.”