As I understand it, in the original bible passage, Jesus says to Peter "And I tell you that you are Petros, and on this petra I will build my church, and the gates of Hades will not overcome it"
And in the original Aramaic, the word used is also a noun which translates as "Rock" or "Stone".
According to wikipedia Petros had not previously been used as a name, but in the Greek-speaking world it became a popular Christian name, after the tradition of Peter's prominence in the early Christian church had been established.
In many Latin languages, Petros is rendered as a the word for rock/stone in that language. For example, French Pierre is both the name of the saint and a word for stone. In others, the gender changes (a rock is female and inflects at the end differently to the male name) but the relationship between the words is transparent.
So how did stone come to be rendered in English as the opaque Peter when referring to person, Peter. It's obviously a cognate with the greek and latin words, but how did it enter the language and end up with its current spelling and pronunciation?
My best guess is that is travelled as a given name across the channel before the bible was commonly read in English, as an Anglicised form of the Romance language name, and when the Bible was translated, it made sense to the translators to use the existing English name. But this is just my guess.