There's a midrash from Otzar Ha'midrashim (printed in 1913 I believe) about the supposed origins of the man who Christians came to call Peter. I couldn't find an English translation, so I tried to do it myself. Hope it didn't come out too bad.
"And it came to be that the argument between the Christians and the Jews grew stronger, for when a Christian saw a Jew he killed him, and the problem grew worse over the course of the next 30 years. And the Christians grew in numbers of thousands and tens of thousands, and they prevented the Israelites from making pilgrimage, and this was a great misfortune in Israel as the day that the [golden] calf was made, and they did not know what to do. Even so, their faith grew stronger.
And 12 men went out to twelve kingdoms and made their prophecies, and the Israelites wrongfully followed them, and these were great and well-known people, and they strengthened the faith in Jesus as they claimed to be his apostles, and many of Israel gathered to them.
And the sages saw this evil thing, and they were greatly saddened, and they said to each other:
“Woe is us that in our times this evil thing has happened in Israel, unheard of in the days of our fathers.”
And they were greatly saddened, and they sat and cried and lifted their eyes to the heavens and said:
“Please, Hashem, God of the Heavens, give us advice on what to do […(not completely sure how to translate the rest of the prayer)]”
And when they finished, stood a sage from the sages and his name was Shimon Kippah [Simon of the dome], and he had used [in the past] the Heavenly Voice, and he said to them:
“Listen my brethren and nation, if what I say be well in your eyes, I shall separate these people from the congregation of Israel, and they shall have no part in Israel, but only if you shall accept this sin upon yourselves.”
And they all answered in unison:
“We shall accept this sin as you have said.”
And Shimon Kippah went into the hekhal [part of the temple] and he wrote the Great Name, and cut his flesh and inserted the Name inside himself, and he left the temple, and he took out the Name and learned it, and he went to a great city of the Christians and he called out:
“Whosoever believes in Jesus shall come to me, for I am his apostle.”
And they said to him:
“Give a sign and a miracle.”
And he said to them:
“What do you ask of me?”
And they said:
“The miracles that Jesus did during his lifetime, do as well.”
And he said:
“Bring me a lepper.”
And they brought him, and he placed his hands upon him, and lo, he was healed.
And he said to them:
“Bring me a dead person.”
And they brought one to him, and he placed his hands upon him, and he lived and he stood on his legs.
And the people saw these, and they fell before him, and said to him:
“Indeed, you are the apostle of Jesus for he did these same things during his lifetime.”
And Shimon Kippah said to them:
“I am the apostle of Jesus and he commanded me to come to you. Swear to me that you shall do all as I command you.”
And they all said in unison:
“All that you shall command us we shall do.”
And Shimon Kippah said to them:
“Know that Jesus was a great hater of Israel and their Torah, as prophesied by Isaiah: “Your new moons and your appointed feasts my soul hateth”, and know as well, that he is not wanting of Israel as prophesied by Hosea “For you are not my nation”, and though he has the strength to uproot them from this world in an instance, he does not wish to annihilate them, but he wants to leave them so his hanging and stoning be remembered forever. And all the torture he went through to save you from Hell.
And now he warns you and commands you:
To never again harm a Jew, and if a Jew would request a Christian to walk with him a mile, he shall walk with him two miles, and if a Jew shall hit him on the left cheek he shall turn his right cheek as well, in order for them to earn their reward in this world and in the next world they shall be sent to Hell – and if you shall do all of these, you shall be rewarded by sitting with him.
And he commands you:
That you shall not keep the Passover but shall celebrate the day of his death, and instead of the Festival of Shavuot you shall celebrate the fortieth day of his stoning and rising to the Heavens after, and instead of the Festival of Sukkot you shall celebrate his day of birth and on the eighth day of his birth you shall celebrate the day of his circumcision.”
And they all said in unison:
“All that you have said we shall do, but only if you were to remain with us.”
And he said:
“I shall sit amongst you, but only if you shall do as he commanded upon me, to only eat weak bread and water, and you must build me a tower inside the city where I shall remain until my death.”
And they said:
“As you said it shall be done.”
And they built him a tower, and the tower was his living quarters, and every day they supplied him with bread and water until the day of his death, and he sat inside [the tower], and he served the God of our forefathers, Abraham, Isaac and Jacob, and he created a great many hymns, and he sent these to all of Israel to be remembered by them throughout the ages, and all his hymns he sent to his rabbis.
And Shimon sat in the tower for six years, and he died, and he commanded that he be buried in the tower, and this was done. After this, they built a grand building over this, and this tower remains in Rome, and they call him Peter, which is a name of a rock, for he sat upon the rock until his dying day."
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Over the generations, there have been those that hold that this midrash - or some version of it - is historically true. Rabbeinu Tam, one of Rashi's grandchildren held that not only this was true, but one of the hymns that Simon/Peter wrote during his time with the Christians was none other than "Nishmat Kol Chai", which Jews say every Shabbat and Yom Tov during morning prayers. Others have voiced harsh disagreements, of course, so the midrash has become quite controversial.
I'd be interested to hear what people think of this midrash.
