John Mark was a travelling companion of Paul and Barnabus (and possibly Barnabus’ cousin), and a man named Mark was also a close companion of Paul. Interestingly, when Peter escapes from prison in Acts 12, he flees to John Mark’s mother’s house, which was an important gathering place for the early Christians. So Peter and John Mark likely crossed paths, and it wouldn’t be surprising if they developed a lasting relationship as a result.
All that said, Mark, the traditional author of the Gospel of Mark was not one of the Twelve, and probably wasn’t an eyewitness of Jesus’ ministry, either. According to tradition, Mark’s gospel is based on Peter’s account of his time with Jesus—which could be why Matthew and Luke’s gospels appear to be based on the Gospel of Mark, even though Matthew was an eyewitness!