My new local monastery is called the Abbaye Sainte-Marie de la Pierre qui Vire. The "Pierre" in the name has nothing to do with St. Peter, it's a literal stone at the site of the monastery. It's a boulder balanced on top of another boulder in such a way that it can be turned around by hand ("pierre qui vire" means "turning stone"). That's the old story, anyway. Such instability made the first monks there so uncomfortable that the turning stone was cemented to the boulder beneath, and a statue of the Blessed Virgin Mary fixed firmly on top. They kept the name, though, even when the stone no longer turned.
There is something in common with St. Peter after all. Jesus gave him the name, which means "rock," as related in Matthew's gospel: