You tell your friend that being first doesn’t always equate to being the most important. Also let your friend know that it wasn’t until the Church spread to Antioch a few years later that these Jews who followed Christ began to call themselves Christians. Would the Church of Antioch not be the first Christian Church since that’s where we first started calling ourselves Christian? As an aside, St. Peter happened to be the first bishop of the Church at Antioch.
As for early church documents referring to the primacy of Rome, look no further than to St. Ignatius of Antioch. St. Ignatius was the third ever bishop of the church at Antioch and was martyred somewhere between 107-110 AD. More importantly, St. Ignatius was a disciple of St. John and church tradition holds that Ignatius was ordained a bishop by St. Peter himself. Anyways…while St. Ignatius was on his way to Rome to be killed in the Coliseum he wrote seven letters to seven different churches. We learn a lot from these letters, particularly his views on the importance of the Eucharist as well as the first ever recorded usage of the term “Catholic Church.” In his letter to the Roman Church, St. Ignatius is in awe of these guys.
He writes: “You [the church at Rome] have envied no one, but others you have taught. I desire only that what you have enjoined in your instructions may remain in force.” He also has this to say: “…to the church also which holds the presidency, in the location of the country of the Romans, worthy of God, worthy of honor, worthy of blessing, worthy of praise, worthy of success, worthy of sanctification, and, because you hold the presidency in love, named after Christ and named after the Father.”
Also point your Protestant friend to the Letter of Pope St. Clement I to the Corinthians. This is the earliest document we have of a Bishop of Rome asserting his authority over another church. And please keep in mind this letter was written in the final decade of the 1st Century. So the Bishop of Rome had already begun to see itself above the others before the dawn of the 2nd Century. As the historian Sir Nicholas Cheetham had to say: “Clement asserted the primacy of the Roman church in no uncertain tones when he rebuked the Christians of Corinth for lapsing into dissidence and schism. The Epistle chiding them for their quarrels and usually attributed to him has the authentic papal ring, both authoritative and paternal.”
So there you have it.